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.4: 1 — A First Course in Probability: S. Ross — Maxwell Macmillan — 1989 . -... ?? 2 A,
2 — An Introduction to Probability and Its Applications : Larson & Marx — prentice 1 /4? Hall — 1985 . 4,
/.< .k; 3— Modern Probability Theory and Its Applications : Parzen — John Wiley 1992. c..<
/2 4 N N ., 4— Probability ; An Introduction : G. Grimmett and D. Welsh — Oxford — 1991. 4) e/ ,q
> 5 - Probability and Mathematical Statistics ; An Introduction : L. Lukacs —
Academic Press — New York — 1972. N N N
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r-- .,,....F7.1.5MZMVINZes‘LUMZYCZfrigkItVi=f127.ErgfZ.SifriUriM701-55:ZainalgitVgal=6:1=glintl!MweZiZZIV"-", *Nefgati127-2AITZFATZ:=IMIEL+1.'1%1112r46;11i=3;TOW.Z111:::aa st!il
2 Chapter One -: (.3.0 LI,..aill
2 Probability
2 Definitions :k4--A-J ,---iy..)15,3 1-1
16 2-1 Conditional Probability L.11--*3- 11
18 Independent Events:;‘1'41,..11 'LL...1.)i j..11 3-1
21 Exercises :autAs 5-1
24 Chapter Two -: 4.5:412 (.3 11
24 4-1\_9.) Random Variable (RV) : u•---,,j
24 kc.1_03 L_.ku,Ls 2_2 (.5.1,1i,t,t1 j.!.;•;A
25 3-2 Discrete RVs Siji
33 4---1_9& \ti 4-2 Continuous RVs . c.i ."':_, Li-L,'14.\\ L.;Ilyt'l l ..),!;1 11:
38 Distribution Function : L-* 5-' .,0_3,11:k.11.) 5-2
46 Moments :,..33,-11 6-2
50 Moment Generating Function : c._93.31 43:; "A-11-) 2-6-2
64 Exercises :aut-.3 7-2
72 Chapter Three -: L-'1' 1V-In (.14-aill
72 Discrete Probability Distributions :
72 Bernoulli Distribution : c,-1_93_):?-.1_33.3 1 — 3
76 Binomial Distribution : C.):-1.- 43 L-7),._51-, 2 — 3 i
83 Binomial Distribution Negative :,-,L-.1 L:),?-1- 43 -,?._3_5:1 3 — 3
88 : \\611 -:?_)_)D 4 — 3 Geometric Distribution ,i,-,
92 Poisson Distribution :L19-1.5: LY:).3.-; 5 — 3
101 109
Hypergeometric Distribution : ,--.)\all c:9Ja L.,0.3:111 6 — 3 Exercises :auk-4:3 7 — 3
111 3,42111 Chapter Four - : :11..)11
111 Continuous Probability Distributions :',;yz,--Ali 1 /44L-s..'D c-.31-,y.35111
111 : 1 — 4 Continuous Uniform Distribution J-41,-(31 ,-163-1 \-,..).5:311
119 Exponential Distribution :L.5-,-.i tu5.-)11 2 — 4
126 :--4,1-31 :)._93.11 3 — 4 Normal Distribution
130 Standard Normal Distribution : ,-,41 ce-4-11 t:0_9ali 4 — 4
135 Gamma Distribution :L.1- L-:.,3_5.; 5 — 4
141 Chi-Square Distribution :41c -*_)-4 -:?..)3:1 6 —4
143 Beta Distribution :k:;:u.. t:1_33:3 7 —4
147 Exercises :aut-,L 8 — 4
: • :
150 Chapter Five - :
150 Joint Distributions 150 Bivariate Distribution L5: 11,A11.-".11 t?...)_":111 2 — 5 152 Joint Distribution Function : 11-CJt'',4 :U1 - 2-2 — 5 t,1_)_,:th 153 Marginal Distributions :(1-k42-41-i-027.1i..911 3-2 — 5 155 Joint Mathematical Expectation : LJ-1_5-11,-.-1.1,1J.1 3 —5 t:s._3:11 160 Covariance :_.):1-;.:D 4— 5 162 Correlation Coefficient :-1.1-.65fl L3-41-,-4 5 - 5
164 Joint Moment Generating Function :..9_5.-11 4_511 ;:k-__;";-,11 -;k313 6 -5 169 Exercises :(:_)4J1-4:1 7 — 5
170 Chapter Six - -,-,113 111 170 Conditional Distribution 171 Conditional Probability Function : .:YL...)L1.11.‘41.1-A1,NI:k-Lil 2 — 6 173 Conditional Distribution Function :;Y.L.)-di-litaiill az 3 — 6
176 Conditional Expectation :co-1= 1* 1 &..3.-111 4-6
: 180 : 1.k:'.L.)-j;321 .‘2.1 S-113-a I.L11-111 5 —6
Moment Generating Function for Conditional Distributions i 182 Exercises :C.):.u1-.:3 6 — 6
; 183 Chapter Seven - :-.11-,,,11ci,Aill
183 -:
Independence of Variables (Stochastic Independence) 184 L:liu.!;1,11 ZA,ii_. 3 — 7 LI', ..AiL..1 L:).. c..lil
e 192 Exercises :c.),L)-4:, 4 — 7 e 7
193 Chapter Eight - :&4111 c.1.,,aill K , 193
411_921,-11 6:11..),311 -:(')11) Distributions of Functions of Random Variables (Transformations):-
v 193 Discrete Variables _: ;0,-;•,.11LLIIJ.;;1-11: Slji 1 — 8
193 The -:-.) :.1-11-.. L3,1...91-1 — 8 case of one variable _.):. ,3' L.53 -,LT 201 The two :C.).21 -;‘.. al_.)4;.1..I-11-.. L3-,:311 2-1 — 8 case of variables (.53
Y
"$ 205 Continuous Variables -: :W-4-11 '=11.).:'1 11 : :,U 2 — 8
207 The -:-1- 1 .3..)4;L:,-* ::k31-,. J71_9- 11 2-2 — 8 case of one variable (.53 A
210 The C.),u.2". :k-11-,. (-3:1_9- 3.1 3-2 — 8 case of two variables: gii , v 215 Exercises :J1-z 7 — 8
217 Problems '1 c•Jt.B.1 ;t13.,u1 j ;k..a. 3.11... ;e
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: L:j1 4:1
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13 0:14 L.Z1S.1
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P(AB) = P(A)— P(AB)
P(A B)= P(A)± P(B)— P(AB)
B A .6
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L71,-"J E{-1214i 1-44° -149 C14 •
P (A73 u AB) = P (AT3) + P(74B)4)4-Az-
= y2 P(B)— P(AB)= 1 112+2/3-Y6 =7/2
;41111 .1i1:14 I ()S....2. :4.1LA
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1.4 C. • • •
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P(Au B) = y4 , P(B) = 2, P (A) = X : B 3 A 13lS 13) .1
1)P(AB), 2)P(AB), 3)P(AB), 4)P(Au B)
P(Au B) .‘ , o -ti P(73) = 0.7, P(AT3)= 0.2 L:-.1:31. 13)
P(A)= 0.3, P(B)= 0.45*L-.Zik-_9 LAIS.1.11,2 B 3 A
1)P(A B), 2)P(7-1u11), 3)P(AB)
P(A) = 1 - P(A) = 1 - y8 = X
1)P(AB)= P(A)+ P(B) - P(A B)
2)P(AB) = P(B) - P(AB) = Y2- X= X
= P(A B)= 1 - P(A B) = 1 - -3 = 4 4
4)P(74u -A)= P(AB) =1- P(AB)=1- Y8 = 7/8
01
P(71 u73) = P(A) + P(B)- P(AB)
P(A u B) = P(A)+ P(B-A-)= P(B)+ P(A B) = 0.3+0.2 = 0.5
- 13 -
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1)P(Au B)= P(A)+ P(B) = 0.3 + 0.45 = 0.75
2)P(74 B) = P(AB) =1- P(AB) =1- 0 = 1
3)P(7413)= P(B)- P(AB)= P(B) = 0.45
qua.A.11 L5ALL9 Lealin ;LIIA.11 :14.1li
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HHH HHT
HTF
THH THT
TTH TTT
r
Li
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= {HHH ,HHT ,HTH ,HTT ,THH ,THT ,TTH ,TTT}
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n(A) 1 P(A) = P({H111-1}) = =
n(f2) 8
P(B) ‘,1311,411
B = {HI-IT , HTH n(B) = 3 n(B) 3
P(B) = n() 8
C = {HTT ,THT ,TT.H} = n(C)= 3 n(C) 3
P(C) = = n(f2) 8
:(5-1) dtiQ
F LJ1h1 L.,55LZI" M J,C.111 L....*LAS 14i Lai terial ';‘111.c..
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Q = { MM , FM, PT} = n(Q) = 4
a4:4L} =A kl.L,31 I
P(A) Pet .M.A41)=
{ Lti'i Lslc, = B
P(B)= P({MF, FM ,FF})= —3 4
(Y)
- 15 -
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:(6-1) Lrii4
Ls_tc. cji 1 ..5AU 6 -;c1 -51..
{ LOP (5Ic Siy,n j_94..1; }= A ul 01:1;1'1
A
n(1-2) = 2" =26 =64 P(A) = 1 64
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164 %
kwi‘loyatit Lil0j.›.11I 2-1 (Conditional Probability'
LislA (1-2-1)
JJ .)41 t-340-.1 —()1-211. J-12-4.11
A={)1 L63.4} L.,./e A Af'.1.t__,.11 L 1 Lfi UJ 4:1A
sL . 01..11 cjiU JIA:=1 i B=Ik1I4 '&44 ;1-,,,-,11) Ls-it B
• P(A I B) . 11C13 B 1.41a A -1 Lsi. _Al 1 J1-4141
C.).4 (J.11:1 -LW .t! ) 6111.011 P(A) cy2 j451 P(41 B) late laIlfiA L.53
) 2.4 t jk, 2 Zin..1.31)..2 4,a31 (: ji J1.1.31
J 4.0 (.5:31:1124 Lsi&I
• Ls-kJ...All
1A-.1411 J1-4".1z.' (...1.)pLi (2-2-1)
LJ LaiLij B LIc A I (:).5131., B j A c:J.31.S 131
:411:1- 11 p (Ai B) .3.o.yj A _I L,J.._,L1 1 L.11-.1- 1-.)
- 16 -
Lr
-1- L3Lc.a.A J..31 ,
P(A1B)= P(AB) , P(B) > 0 P(B)
4i ju3 1-.13
P(B I A) = P(AB)
P(A) >0 P(A)
:(7-1) dei4
L 3 ± Lcic. c,53
L334.11 (.53 &tall
(A) ..L.4,...1 (A) L,..b....,
(B) .1:,5.J 5 12 17
(73) ''''". 9 4 13
14 16 30
•JI,•;,1 4c‘..r.11pla
C.C.A.D.1.11 b.0.0 CJA Di 0
5:JAI .3 j„.1a..1 L.-L-GLA k3t.? L1C- r...0t3_}.111 &a.c3 LDA (
={(...3-..5)4.-.1 i t a -11 c.-1} n(71) = 16 ::+411:111,Lo1,...1
n(A)=l 4
n(B)= 13
B={_31 €7'11 IL'AUJ 1 n(B) = 1 7
:P(B)
- 17 -
-1- (.31-4:-.,
n(B) 17 P(B) = = — = 0.57
n(Q) 30
:P(B IA)
P •
(AB) n(AB)/n(Q) n(AB) 5 P (B I A) = = 0.357
P (A) n(A)/n) n(A) 14
(y) 'Ojai L:941.11 4.311.11 csi VI
A L631 ai (30) 0 ;1 /44,11c.1_,.....i L,. A tjk,Il :k1)1.11.4
cL A 1:il .A LP,ILL B ji412 j (14)
.P(B IA) P(B) 0.44.11a)1:1.-.1taia..V
c,t).4.11S.11:9 (3-2-1)
(Multiplication Rule)
P(A I B) = P (AB)
P(AB) = P(A I B).P(B) : Ls192.11JI.z.1..s/ P(B)
P(B I A) = P (AB) P (AB) = P(B I A).P (A) P(A)
:
P (AB) = P (A I B).P (B)
= P(B I A).P (A)
.1 y>,..12 L3.2lz.SJi laj4c. ,J113.13
:k ret C:1Apll (3-1)
(Independent Events)
31 ,L1.3.1.. Lcs ut_c 131 Citri.w. L.4.3i 13 A ail3.1.z. cf r
.L5J&C11
0
Li
- 18 -
-1- 215 .3,11 (D1-.4_59.11,yziLi 14=-4
: 4.11-1
j4,1 13) Ly ,<-114.9 B L:fr ii4 A Zi.A..11 L.) (51° I c).
t.1.11 -'4a111 J.3.)11- I
1) P(A I B) = P(A) 2) P(B I A) = P(B) 3) P(AB)= P(A).P(B)
)5toWl -1
j)1.1w,- LtV1311 IL>L11
:(1 - 2 - 1 ) Z4)
DIA B 3 A DtfsiL,11 callS
.LD 73 3 71 - 3 iptiliw* B 7/111 -2
J.2,111: iB jA ( -it' 31.11 La$3 .1
P(AB) = P(A).P(B)
L.I-I.S = P(A-13-)= P(A)- P(AB) = P(A)- P(A).P(B) = P(A)(1- P(B))
= P(A)P(B) = R.H .S
.L411-.11‘ 4.13_51.4. .2
.3 P(AB) = P(7 -4).P(B)
.S = P(AB) = P(A B) =1 - P(A B) =1- {P(A) + P(B)- P(A).P(B)} =1 - P(A)- P(B)-1- P(A).P(B)
= P(11)- P(B)[1- P(A)]
= P(74)- P(B).P(A) = P(A-)(1- P(13))
= P(A).P(B) = R.H.S
Li
Li
Li
- 19 -
n\ (1)
1 \r )
(n"
n — r =n &
n\ n\ =1 .
\n ) I
( n in+1\
r —1
( n (3)
n-1" n\ (2) n
r —1 ) =r
-1-J-
:U1 ji=
:
Lax= Ci B A Lf_14,z. (:)c, Ji
P (A n B n C) = P(A).P (B).P (C) .1
:
. J.5,4aM csi A2y.c. c.LC3i = us,,1
1-1.41.?.71 . ola 6:4 . A = {1,2} , B = {1,3} , C = 11,41
E+.141.4c;J' u:au (4-1)
0
0
(n\ + 1 = (4)
1 = n +1
( n\
(5) ( n _n—r+1
r —1 )
- 20 -
-1- u..==.1 215 1,J5 IA asz, .Laz.A •
L*J;Iii-A5 (5-1)
. (4-1) utt9,1 (1
P (AB ulAB)= P(A) + P(B)-2P(AB)
• ..k-,2.1ziA, r4 J J a9C1 -1:q3 t..5:i3 ) (3
j_941, J.1 () ;k.0.41 041 i (i)
L5L.A 0.41 çjL di% (a;). CLN ..Yr3
f
L -;41t,•11 .41191,31 jt.:111
JS 1 (4
k.05111 L:..k1).411 (y) . H ulc. --.11
• ;1-b..) cL11.) 3 ct:=2:•14 ‘L- b-C 5
P(A B) = 0.4 , P(Au Bc)= 0.8 i3) (5
45L9 jg .;k3514 E. ;LIM t1,1-11 (6
: J1.13..1 43ti b i . H
b9-1-='11 J.96 1;
• by-.Jy (L,I)
•11111 DI 0)
:;k411-311 c=1Vt.-11JS Lci P(B) 4itiP(A u B)= 0.8 P(A)=0.3 DIS 13) ( 7
. A c B (c)cjtArix. B A (,-;-) ut-L31. B A (1)
- 21 -
1.3z..43
C j D deri P(C D) = P(D C) = P(C) = 3 ulS 13) (8
C 3 D
. ‘113
P (A) = 0.45 , P (B) = 0.35 , P(A B) = 0.57 :citS (9
:C13
P(B. I A) it —1 0.35 B) 0.44 C) 0.57 D) 0.74 A)
PP IA) (.31-41=.V1 -2
0.274 B) 0.64
C) 0.455 DI_ Al 0.35
P(AuB) z,-..\/1 _3
0.60 B) 0.80
C) 0.45 1:).1 0.20 A)
A , B Lill (10
ut:Lt.. A , B Lhil -4
A) AuB----0 B)AcB C)AnB=0 D)A=S,B=S
: LDIS j111-41,,, A , B
Q
- 22 -
L..)1 215 -juS3A
A) Au B = 0 B)P(Au B)= P(A)+ P(B) C) Ar = 0
D) P(A fl B)= P(A)P(B)
:;k41. clati_;6,31 Lga,.)
A) P(A n B)= P(A)- P(A n B) B)P(Au B)= P(A)+ P(B)
C)P(A n B) = P(A)P(B) D)P(A - B)= P(A)- P(B)
3 .0.3 jib c,1_,..at41 j1.4iLl 44.9 J1-4:11 DISE)) (11
0.12 J1-13-T LU-NI -6
B) 0.3 C) 0.21 _p_). A) 0.18
JI-f-';•T Lb-A a'-'1 Y_Y".13 YYJ'j 3-411 -7 0.12 B) 0.3 C) 0.28 D) A) 0.18
m P(A 13c ) = 0.2 C.J1- C1:123. Eli 04.43 allk.)2-4 13 _5A Lig331---11cL131-S13) (12
P(Ac nB)= 0.1 j P(A n B) = 0.3
LI A) 0.20 B) 0.40 C) 0.36
P(Ac ni3c) ,_
D) 0.55
P(Ac u Bc )= (2
A) 0.5 B) 1.0 C)0.7 D) 0.4
: P(AB) = (3
A) 0.85 B) 0.45 C) 0.65 1)) 0.75 : B 3A DUli-,-11 (4
A) i3141,„„ B) C) ut4 - D) LI)1:1L9L:3-c. a:1%1,A
P(B1A)-,-- (5
E) 0.8
F) 0.4
G) 0.7 H) 0.6
- 23 -
F
215 -1-
($It Lotaiall
kithaj9
(Random Variable and its Functions),
:LAA (1-2)
L..1 1.41.4.1&H L 13:12-. 1.,••=2i _))..42111L4 13.41.4:11. 01
cy11.4.4 ;I jai, Lb.). .31*A.34. J.C1=•3 (5.:111 .11C. ith
L.6.31 1•4•1:la
4.1147 / 13A
L:).4 ji j 4. ajx-Alt ;1;ati _.›L.4,%1.11P-4 11141),Q Nr• ,.:111.4111
4-1-IJA3 11 ;‘;, —11 c.-13. ;1131=-. e4i.Le, 63A j&I
11114.
LL. 11).4 LLL.V.1 11.1.4G. •fla H a*); L:11_1).4 .1.1a • 1
• 4...11C 4) .2
u-A .t 100 e,.A 160 03.3.1. ii (:).4:111 .Lac. .3
.4411.0,411 J .4
.4.. (52, j11.5311sji
Z Y 31 X _4 -6 jAig. 4.311.3 j1;1-11 L1523 Di t_013:1.41
Random Variable :0-11...3,14 ..)71.kisali (2-2)
(1)
uu.',1 X 1"Ln 41 Cb . 14)4 rqa ptiesi clh 1:11
.5
TT1 Li
3
- 24 -
0.1215 •;,)-cL. -1- L.IL-1
1]
F-1
4i:143,1c. .j4i:14 (we SI)Q L w X(w)
.:t R X : --> X(w) R
.( Z11,) X L5:11i.t.11 J.2;1.11 C.11 )
:„.11.9.412.• J.:2151. u:a2i (1-2-2)
c UJ 0c--- L L.J1...41_52;c. LD1 Yi X DIS
C E R
i) (X + Y)(w) = X (w) + Y (w) ii) (X + c)(w) = X (w)+ c iii) (cX)(w) = c(X(w))
.( (0.1b x JL L,.51 L•ji )
Random Variable :4114,24:L3 c..ih..4.4.1* e9.1i ( 2-2-2 )
14.43.u ü- LS (2)„
U L59 (Li) LILA
C_Jr- ) 1J-41-4 1.):.;.14L5 —.!
:(Discrete Random Variable) 1.12:2-1A1' I (.5.11j2210 J4.114.1' (3-2)
:4-1.4,6 (1-3-2)
13) (J.L,ada X
:(1-2)
:;i411:111c_J•b4;1,11':uS,...11 c,111.1.9`1
- 25 -
2/3, 1/3, 1/3, 0/3
_1_ 215 s fj..p....! _3;11 c:4)111.)J-cit.;
;k4-6 L Y .2 -12a C.S.'=4:4X .1
- diet! Z .3
li.041.41:=1:14.11
f2 = {HHH , HHT , H'TH , H'TT ,THH ,THT , 1 1 H ,TT7'} Li
X= 3, 2, 2, 1,
Y= 3/3, 2/3, 2/3, 1/3,
Z= 0-3, 1-2, 1-2, 2-1, 1-2, 2-1, 2-1, 3-0
= -3, -1, -1, 1, -1, 1, 1, 3
X : ---> {0,1,2,3} c R :.g{0,1 ,2 ,3} :„..0,3 X Ult.a&t,l, Liall 4.10i c.,..;.13
‘1133
X (HHH) = 3 X (HHT) = X (H'TH) = X (THH) = 2
X (THT) = X (HTT) = X (TTH) = 1
X(177) = 0
Y:f2-->{0,Y3 ,2/3 ,1}cR :(141143
Y (HHH) = =1
Y (HHT) = Y (HTH) = Y(THH) = 2A
Y(HTT) = Y (THT) = Y (TTH) =
Y (TTT) = = 0
Z : {-3,-1,1,3} c R :j trIS3
• rl
- 26 -
-1- Jul-1 u---1 215 -b.)c•-. ,971 J.4.4:1U 1414
Z(HHH) = 0 -3 = -3
Z(HT7') = Z(HTH) = Z(THH) =1- 2 = -1 Z(HTT) = Z(THT) = Z(TTH) = 2 - 1 =1
Z(T7T) = 3 - 0 = 3
: a L516 X : -> R HHH --> 3
HHT PITH 2
THH X : = R
1111 THT 1 TTH TTT ---> 0
.Q ç.Lc..4.3.? X (1)
0
x
(2)
• Z yL1)1.).!;14
0
:(2-2) del-4
Eli cLAY.ia c x.k _53,sfr 1 H 11 j9$13 "
x 4_9i
LI :L5A (I))
= {//, 711 ,7111 ,1771-1 ,777TH , X = 1,2,3,4,5,
X(w) = {1,2,3,4, : ccto (X :k 4,..,11 :,-,41) X
0
L.J
0
- 27 -
-1- (..31-4al
:(Probabilitv Mass Function) ;L411.4:111 :kla.C11 Z114 (2-3-2)
;CL5.1' 1 Z11.1
1+1 Z2..1 :4-11<11 :1 /4.1L 413z,ti x 131
1.A.0 .f(x) j fx(x)
f x (x) = f P(X = x ;) , 1 0
X = X X 1 2
0.W
LsAtAa4
(Discrete Probability Distribution Function)
:a4.1. f(x)
(i) f(x) 0, V x , ( ii) E f(x) =1 vx
.f(x) :t..11.1.L41ktS11:k.IL
j.±;1.1 1 :LL J47:3 als4.-.1.21 • u44)..fali L 231.) 419
J.S1 ih1.1,31 LIT_, X j.!;1.11 r4i pa?.
• X t133•411 J.!. '\ L.,i1)
:(i:411)&ILA' II 4.S19.1.1.11 j4LAI' I ;6_1 A.1.4.11 .4.411 (3-3-2)
(The Expected Value of X (Expectation):
Mean A X ,;11_,,t.11 ).;•;',,11 4 JLA1 •4 ,S;11
-E(X) .91 _3414 1+1 X Ji;111
Z,S,j11 f(x 1),f( x 2) , LIN x x 2, .... 41 3&t,, X LDIS 131
X j 11
ii Li
- 28 -
Li
,u = E (X) =Ixf (x)
E1X1 ( X) <00 19.a..1 i...412..4 1.1)11 11LJJ Lpi
.x
u•=1.9.&, (..).11.2%1
Lri.:1t3 jlaiA C 3 (X) .;141., :;1,1:1S11 k113 k11,1,i1.4 '.1.41c* . ..)4;6.4 X Lil.S 1'3)
: c G R
i) E(c) = c
E(c) = cf( x) = cE f (x) = c.1= c
E(cX) = cE(X)
E(cX) = x.f (x) = x/(x) = cE(X) :(plAj411
b j a E(aX ± b) = aE(X)± b (II)j (I) cidai
c9J ( r j4i'LL4' L:c2,31 c5/4 g(x) L:i1,3 X g(x)
E(g(X)) = >g(x)f(x)
:(Variance and Standard Deviation) cp1,42-421 L1_.9 C.)4141 (4-3-2)
:
0-2; .3414 4-1 .)-4J,? 4-1**-1 P 1...3;1° X (:.)\- 13 )
V (X) _91 Var(X)
Li
- 29 -
(x - ,u) 2 f (x) x
=- E(X — ,u) 2
c 2 = V ar(X) =
L. J
0-2 = E(X 2 )— ,u 2 :LEA
u---1215
:j LS &J 9 i L5,43.)A1 ja„31 c4ji.!-11
a -= 0-2 = VVar (X)
•i g .3:14 ap. 4.1i X c ...19ccC.) Lg L'i44111
31.2- ,3 (yi9:3 4.1i cc. j .14X-4611 L-91)=37 (X Lr-11.1 .9 t..D..4) :LT-DA LL'11 .9 csA
LAi (...41 'II•1-till
= Var (a) = 0
. ji Lagl ),,t1 jLi a
: cjIr.j411 Li
a2 E(a — E(a)) 2
= E(a — a) 2 E (0) = 0
Di csl 'L11-.112 JI'° 11 c.s..).!"1 L-.6-2411 471.1 3 Li
c a = \IV (a) = 0
ii) a2x =Var (aX) = a 2Var (X)
1.3
.'. [Tax = VVar(aX) = a 2Var(X) =lakiVar(X) =laic x
: c jI.A.)411
- 30 -
Li F71 Li
-1- jul,I ‘31.7.t,50.11.).cca.:1
o-a2x =E[aX — E(aX)] 2
=E[aX — aE(X)12 = a 2E[X 2 = a2 c4
a ar 'Iala x
o-(2„xti,) =V(aX ±b)=V(aX)= a 2V(X)
••• aax±b= aax = lalax
j..111
V (aX + b) = E[{(aX + b)- E(aX + b)} 21
= E[{aX + b - aE(x)- b} 2 ]
= E[{aX - aE (X)} 2 = a 2 EP( - E(X)12
= a 2V(X)
C ax
:(3-2)JtA
:.1,31 (1-2) L.111.
_9 V(X) _9 E(X) -/-?_91 .2 d 411.4_9 X gsjyt.11 t-i.ag
. V(Y) E(Y) Y=2X-1 citS 13) .3
: ,1 ,2 ,31 X SLi, (1-2) J11. J.-
(1)
(0)= P(X = 0) = P({TTT})= X
L:11) 0,1)0 4.91_,L144
j*(0) = P({17'7'})= P(T).P(T).P(T) = y,..y.).y2=1/8
- 31 -
34. 2i '8
-1- Lit-1- 215 L
:
f(1) = P(X = 1) = P({HTT THT u TTH}) = Y8
OR
= P(HTT) + P(THT) = P(TTH)
1/2.y2.y2+ y2.y2.y2+y2.y2.y2 =y8+x+x,y8
j1-411-.13
f (2) = P(X = 2) = X & f (3) = P(X = 3) = y8
L33.14 LEi X j.!,;1.‘1 Z4tas
X 0 1 2 3
f(x) y8 3/8 X X
f (x) O, V x
ii) (x) =1
:L5311411 (X1,1\
f(x)
Li
1 2 .- X 0
:Lx1 J.3.14 a.11-.643 tk;)31 -44 (2)
- 32 -
ci
-1- 215 "._)-C-1.. ..9-.11 &..)111.),z1U
x 0 1 2 3 t.-9A^11
f(x) X X y8 y8 1
xf(x) 0 /3 8 % X 13/8 =1.5=E(X)
x2f(x) 0 X 1% 9/8 24/i =3=E(X2)
-1-?ib'lc-1 J34-11 L:)-.
E(X) =Exf (x) = 0 .Y8 +1.X+2.Y8 +3 =1.5
E(X 2 ) = 3 V(X) = E(X 2 )— (E(X)) 2
= 3_(1.5)2 = 0.75 ax = VV(X) = V0.75 =0.866 .
(J71.1A 3 t'4:111 I V(Y) 3 E(Y) .)tkj< .3
E(Y)= E(2X -1) = 2E(X) —1 = 2.(1.5) -1= 2
V(Y)= V(2X -1) =2 2 V(X) = 4.(0.75) = 3 .
(Continuous Random
4_,s13,24-1;LI.4.1' I : 14;tfl (4-2)
:Variable)
)56 ((1-4-2)
•;‘,2*, '2.L12. y,..! 4311 j.!,;-,,11 (L.1..1.' 31) LsJi,t,.11 j ,,,, 11 J.5.1.31 CSI-
Li
[-J - 33 -
LJ - 215 a"z-4 .
:(Probability Density Function)A-LAA--'11 :t.its11 ;',11. (2-4-2)
U. J.,...1. L 32 4' 11
f,.2 11 al 0( I jy.,-,,11 cj... ii:Pat3,4 1l-i4 i
_yo LIU:141 1:26 X _}.4..12-..1 d&.1.1 c:—.11.519.11 (:).4 (3-.311.1. N4qAaH L.)-C•1_9
131 C:att3L9 f(x) L. _3 X Jj g
f(x) ;61-\11 t.4? x 3
P (a X b) = f (x)dx
6i.:1* 13j j f(x) ;61.) 41 4-.41a
.y.111
(1) f(x) 0, V x , (ii) f (x)dx =1
.f(x) LP
(:-.1111-N;l4
131 SL-.%:14 X 45:11.3,12.31 • X .1
(x) = P(X = x) = 0 , V x
x,
P(X = x i ) = P(x l X x l ) = f f (x)dx = 0
Li
:11411311L J L Lcla 4_9L ..,a4J1...znz.71 .2
{a, b], (a, b], [a , b), (a ,b)
-1 -
- 34 -
-1 -
: x Dts 13) .u1
P(aX _b)=P(a<X_b)=P(a_X<b)=P(a<X<b)
.(a,b) l:11+3 AiL:44 _A. Lc:m:1. 13A3
:3A4L.A1* 1_4,,S1,9121* I Lg il42.4.11 Lilja-IM 641413 1:9.4:01 (3-4-2)
U$9:41 u -41.s L..11143 Li---1.11 J.!" L2g1.)- 241_9 ti_9*-111 LJ
:Di 41 .1 LIA1C1 E t-4 _ArvI,11 1.
34 X J. -1-1 1 j2;1,11
= E(X) =1 xf (x)dx
1.a_ L5:9 ocabLil umil 4.13
V (X) = o- = (X - 10 2 f (x)dx
o- x2 = V (X) = E(X 2 ) - (E(X))2
(Tx = +V V (X)
j.!;"14ii L53 ogi:3 <6.3
J.4.411.1
Li
- 35 -
a
• f(x) ;61s11 4 2)LIV-14
0<x<4 8
(i) P(X (ii) P(1 < X 3), (iii) E(X) , (v) a x
f (X) c:13-CS L511.9 J,-=1:3,4 L.J1 13k3 (0,4) b:13.. 1914c• t:11
f(x) csuj.:1 Ilk, (1) f (x). 0 , V x
f(x)
0
4
(2) f f(x)dx = 1 4
/8 11/x
— _ 1/8/
J
- X2
2 )6[1611 1
: 441c:341,1..1 ;61faS 11.11..f(x)
(i) P (X 1) = y 8 x 0 = [f
f(x)
- 36 -
L
fl Lj
34
LJ
'•1
1 cy..4:Lai 215 CJIT) j4.1.
3 2 3
P(1 < X xdx 1
x2 x[92 21 2
4 3
(iii) E(X) = xf (x)dx = fyx2 dx = [)±- 8 0 3 = 2.667
(v) ..• cx x = VV(X) = VE(X 2 )- (E(X))2
4 4 4
E(X2 ) = x 2 f (x)dx = X x3 dx = 1/8 [ 237-1 = 8 0
V(X) = 8 - (-8)2
= 0.889 3
crx = V0.889 = 0.94
:( 5-2) J 114
. f (x) = ax 2 (1 - x) , 0 1
ax (5 V(X) (4 E(X) (3 I'(X > -2 ) (2 a C:14 1.1 ,19 (1
3
Z.,11 kitS li_11.) fix)
1) f(x)dx = 1
a x 2 (1 - x)dx = a [x 2 - o
=a[X
3 _X
4] = ari _ 1= a
3 4 0 L3 12
LI
r73
LL3 - 37 -
cLi" t.,k3L11.y.eaL:
—a =1 =a=12 f (x) =12x2 (1 — x) 0 c 1 12
2) P (X > —2
) = f(x)dx = f1$ 2 — x3- .clx 3
= 12 x — x = [4x3 - 3 4
3 4 = 1y/27= 0A07
3) E (X) = I xf (x)dx
1 r 4 5 1 = 1121X3 - X4 IdX = 12[)---1 = 0.6
4 5 0 0
4) ..• V(X)= E (X ) — (E(X)) 2
1 X
5 X
6
--5-- — -do = 0.4 E(X2 ) =12f x4 (1— x)dx = 12
5) ..• V(X)= 0.4— (0.6)2 = 0.04
crx vvuo 0.2
LeA51;11 ;L114) :.4:tgin :L11.1 (5-2)
(Distribution Function (Cumulative D.F))
:L.a.A...x.1 (1-5-2)
X 4 4k4 4..43
F (x) = P (X x) . F(x) _5-914._,..3.3:111:61.11
f(x) D-4 F(x) 4-4 Yt-i•
f (t) F (x)
f (t)dt
- 38 -
L
Li
11 i
Li
-1- (31-,1-1 215 -;_)C.)-•. C.}44k4 .14-164
f(x) =
F(x) u-. (.:4-1:1S;d1.) f(x) 2t...4:i
{ F (x i ) — F (x i _i ) , i = 1,2,3, • • •
F(x) — I? (x- )
.s..A14. . 041 I di's. .,:i.l. 0 ( ..14 :•, ,.., A x- ,2-.4,..
d J
,„ lx) = r —
dx (x) : ,A1..1AS F(x)(ZIL) f(x) 3
:F(x )
:(SLA1A 1,-.LA X cjI.S F(x) *.j+Z 0..42,11
: "1- 1 3 I .5. 5L9 L:J-°3 .,"11 431'9 JL41 C.).° F(X) .71t-621 .1
Lim F(x)= F(—Do) = 0 & Lim F(x) = F (co) = 1
: L'iL14
F(—co) = P(X —co) = P((1)) = 0 = L51/3 {X < —co}
F(00) = P(X co) = 1 c= 16 {X 5_ co} Zi-,-11 LESS3
• • ti • (J, 3
: 4:1 [0,1] ;jail! F(x)
F : R --> [0 ,11
: x1 < x2 DLS 13) 431 ,(..1)153) F(x) ..k.11.) .2
F(xi ) F(x2 )
- 39 -
-1- JL 215 j 4 J.C.145121J-4-'21-1
{x• x1} c {X x2 } x1 < x2 cjtS 13) :c_ j1.4111 J
P(X xi ) P(X x2) (Llua.
...F(xi)_F(x2 )
:ui E> 0 ill 431 uoa ;11,-ca.c. AR.) F(x) .3
Lim F(x+ E) = F(x) E-0
• L'itit,1 )41
: (:)1 b a c apac• Liel .4
a < b 131 :CJI.4
P (a <X b) F (b) — F (a)
Ei {X __b}={Xa}u{a<X 5.b}
...P(X5.b)=P(X_a)+P(a<b)
F (b) = F (a) + P (a < X b)
P (a < X b) = F (b) — F (a)
LlitEria J.;;1,11 ci_cAz.1
:L.41:111
i) P(a Lc X b) = F (b) — F (a) + f (a)
ii) P (a 5_ X <b) = F (b) — F (a) + f (a) — f (b)
iii) P(a <X <b) = F (b) — F (a) — f (b)
P(X > 1) Li P(x) ti33-111i1 (1-2)3L4 Lci :(6-2)JtiA
J.:;1-1‘ 13, (3-2) Liu-.J.. L:).4
x 0 1 2 3
f(x) y8 X X y8
o
13
Li
- 40 -
,94
tid9:111
F(x) = P (X x) = f (x)
F(0) = P(X 0) = P (X = 0) = f (0) = X
F(1) = P(X 1) = P (X = 0) + P (X =1)
f(0)+ f(1)= 1/8 +Y8 =%
F(2) = P (X 2) = f(0) + f(1) + f(2) = /8 + X = X
F(3) = P (X 3) = f (0) + f(1) + f (2) + f (3)
=X+78+X±X
:L.4.11:111 —11 Ls_lc. F(x)
F(x) =
0, x<0
yy 0<x<1 8 ' -
1<x<2 8 ' 2<x<3
V' 1, 3 x
1. 7/8'
:U1.)
-Function Step
3)
f{ 2)
) 1/8
fo) 0 1 2 3 X
4 4 P(X >1)=1- P(X 1) =1- F(1)=1 - —8 = —8
F(x)= f(x) F(x)
- 41 -
-1- 215 i
f (x i ) = F (xi ) — F(x) f(x) LsIc. j 3 ,11 lv ,C Ji j
: L53,1, LAS
f (0) = F (0) — (0- )=y 8 _o=)
f (1) F (1) — F (1- ) = %— ,=
f(2) = F(2)— F(2- ) = 7/8 % = X
f(3) = F(3) — F(3- ) = 1 — 7/8 = y8
Ai41.11 .:1 -11111 Lyal csk9
. (I) , (ii) 14.4.1Lwi- F(x) (4-2) JItl. cci :(7-2) JVIA
x 1 • F(x) = f (t)dt F(x) = dt =
1 [t2 =
x 2 - — 8 2 0 16
:
F(x) 1.03 g 21<1
0, x < 0
F (x) —x2
0...x<4 16 '
1, 4 x
i) P (X 1) = F (1) =12/6 = X6 ii) P(1 < X 3) = F(3) F(1)
= 32/6 X6 = =
(4-2) Jt. 1.1.1:,344 &l 31t3
tdoll :L11.1 ,61.4 :(8-2) Jti
0, x<0 F (x) =
, x > 0 1 + x
- 42 -
-1- (:)114.5911
.x (:):14 (1
LJ f(x) Lii.91111 a?._9i (2
. P (X a) = 1/2 DIS a 4 ji (3
:3a.11
(x 4.3 ) x 1 + x )
:1_,IaLa 04 X LcIt I 1 1 3 • 1 -t„ F(x) 131
X DIS 11)) c113 Dz. X Lila II)
. f (x) = P (X = x) = 0 V x
f (2) = F(2) — F(2- ) =
0-0 = 0 2 2 =0
1+2 1+2
....)4.;:14
(2
0
f (x) - — (x) - d dx (1 + x).1 - x.1 1
(1 + x)2 (1 + x)2
0, x < 0 f (x) = 1 x 0
Lo + x) 2
f (x) 0, V x (i)
Li - 43 -
(ii) f (x)dx =1
1 dx _ 1.2 th ,
.10 x)2 y
= =_1[0_1].1
y=l+x
rk.11.1f(x) .*.
F(x)
(X)
(3
2 a 1
P (X a) = F (a) = 2a=1+a =a=1 1+a 2
:(9-2) (114
:Ls-45)53* 61.) (:).5111111 u02 LS &AS 131_4.1
a) F(x)= {0 , x < —1
x, —1.x<1 , b) F(x)
0, x<0
yy 0<x<1 4' —
1<x<2 2 ' 1, x 2
:c1.%.11
F(-1) = -1 :cji x=-1 0:1=1 F(x) F(x) cji (a
-Lk 41
- 44 -
c_.) 1 215 (:)171.
: Cji-‘1113.3 ...k .! -41:6 0:lc..1 F(x) (:)..C1 ,(2.4.43)',.l ..± v4111 i.46. F(x) Di 4,L1 (b
1 3 F(1) = — , F(1- ) = 71
2
: 1 i D.
-4;
00-2) JtA
L5iL. F(x) .!ACI 5111 t,03:111 1..))
;L.A.
F(x)
0 , 0.2 , 0.5 ,
x <-2
2 .x<3
x
. (3)=0.25 DI c c2_341Z.0.4 (1)
. X 4-111.yt-11 .).- 1AB LY.14-1113 (2)
.3
P(l< X
P(X > 1.5), P(X = 4)
: c c2-1-.31121 2, (1)
(x) = F (x) — F (x- )
f (3) = F(3)— F(3- )
0.25 = c —0.5 c = 0.75
ja?.i x L„Jy':‘,11 ti5L11
:L531a11 X j4.;t:14.11 L.511,:14 1
LI
- 45 -
x —2 2 3 5 E
f(x) 0.2 0.3 0.25 0.25 1
x f(x) —0.4 0.6 0.75 1.25 2.2
x 2 f (x) 0.8 1.2 2.25 6.25 10.5
E(X) = Exf (x) Li
E(X) = 2.2
E(X 2 ) =10.5
cJ2lSJ5 4 juazU
V(X)= E(X 2 )— E(X) = V(X) =10.5 _(2.2)2 = 5.66
:411111 (1)
P(1< X 2) = F(2) — F(1) = 0.5 —0.2 = 0.3 .
P(X > 1.5) =1— P(X =1—F(1.5)
=1-0.2 = 0.8 .
P(X = 4) = f (4) = 0 .
(Moments) cails.11 :(6-2)
L.k)sti (1-6-2)
. x 3_3.11 A.241.1I • Ls.a5 (uud'al 31) X L;113.,1c.
t1331 N.3.313
:(Central Moments) 64,3s,),41 ,32.31) ii .11.64,41 c),q ,32.21 (1
.r (4.3.cjd411 e3.31) p E((x—) "A
Pr = E (( X — ) r=0, 1, 2,.... 4.1 • Pr
f • I
LJ
- 46 -
-1— 0,==.1 215 bJ A ,5:21 fit.J.5511. 1.y.daU
c.7.771 E(x - f(x) = E ((X — ,u)r )
(x — ,u)r f (x)dx LI
X Z11-, cri
J X a..
• i • 1L,v u
= 1
,u, = 0
11-12 = g2
122 = g(X I-)2 ) = a2
r.:731 Di 41
.4.12:su"Jp.(%)t..11 (2
(Moments about the origin):
• /11,, .:)-0)17k r (DA • E (Xr) :;% Alci \I .L.5,2_,n
i)
ii)
i,3231 1:1,%
= E (X ) = f(x)
f(x)dx
p: = E(X' ) Ul
6-1011:1 (...1-41S:131 ty"."11 ) E(IX1r )< 00 : 102;y
• .1 1„,.%7 • u
- 47 -
•
=1
ii) = E(X)= ,u
11'2 = E(X 2 ) = ,u 2 + o-2 0.2 "1 2
:1-62"J al-4-32-11 AiN-2-11 CD-C-°:1 pi,. pr au...Ak)6-11
1) /1;
2) 112 = /.1 /12 = 0_2
3) ,u3 = -3/4/4 +2i3
4) p4 = -4/114 +6112 /4 -414
k r cjI.Sj E (Xk) 11 LIS ai E(Xr)
:(11-2) dta
yi Aats- 1.)„,.;a4- x 131
f(x)=1 x=1,2,3,4.
.L.5.2A).231 tf-11t131 cz3231.3 j,-amCaJJ L53.111 't4sit.111 izi.3x31
:c 11
= E(X) =ix, f (x.) = —1r12 +22 , + 3 2 + 3 1 30
i=1
=-1.311 1 +2,- +3- +4- ]= 100—=10 r 10 10
1 [14 , el "4 4 +3 +44 = =35,4 354 10 10
_ .
10 10
/4=
= E(X 3 ) = f (x i ) =
E(X 2 ) = f (x,)
C 1
- 48 -
-1- 3-.1-1 215 _9;3 DL.4,11.11).4.41.;
Ir =Eqx-pn=E(x-P)Tf(x) • 113 = E((X — ,u) 3 ) = E(X 3 )-3(E(X).E(X 2)+2(E(x)) 3
= —3,ug +2,u 3
113 = E ((X — 3) 3 ) = 35.4 — (3 x 3 x 10) + (2 x 33) = 35.4 — 90 + 54 = —0.6
,
:(12-2) JA
: S k.Lt.4 f11.. X
f (x) = 2x , 0<x<1
Ls3_5 1 kfi)ein c%_53.131 4
= E(X) = f xr f (x)dx
= E(X) = xf (x)dx = 2x 2 dx = 2 0
3
2 3 o 3
r-71 Li
4
= E(X) = 5 2x 3 dx = 2 x
- I = ()
5
= E(X 3 )= 2x 4 dx = 2 =
.4) (L..53S_),11 e.3.11 l(.2 c.3.131
:( 11-4) ki-1)031
07
Pr E((X - ) = (x /Or f (x.)dx ..A..a.,11 D.4
- 49 -
..J 14s.:
,u3
1 = 21. x(x 3
o —3.-
2x2 327
0 1
+2.(2.`3 )dx= 2i[x4 \ 3) o
—2x3 +-16 x]dx
, +16 x- =2. —1 = —1
= 2[x55 —2 x x4
4 27 . 2 j0 270 135
:;41111;k1,11
cj..,
P3 = g(X P)3 = — 3 PP; + 2 P 3 )3 —1
(24 24)3 ) E((X = 2/5 (3 x x Y2)-E 2 =-1----35
u1.1 (2-6-2) (Moment Generation Function)
cs.1) cta.:1* (14:11,aval L:_11›.;;1,11 c _332.11
I N3.13 Za13.411 cj1 :-.L...1.t.‘31 C.1)=114 N,3,..11
J.:41'11 c2_13-%31 Ls-la (-1_3--11 1.:L•11 c;•11°
:Li ** (1)
. e I I a all .411 A.AA I Li% (ILL.") j.11 ji) X Ls:3191,3i
:crit Mx (t) r-33.3.1 3J..4i :61.111 (Di .M (t) j M x (t) 3414 lei ..5.4_yj
M x (t) = E (e ) — co < t < 00
431:131.c :OLD
le' f (x) M(t) = E(e xt ) = {f e xt f (x)dx
X
ri
- 50 -
-1— ciL4-1 0-gzti 215
1.)/401.1 441.1.11 .4
:1%432-11 (2)
L-71.-4 D.41)-12 LS.14.1 (j LA.P• 4-9.3) X .J.:4-'11 (sic"
LoS -;41:1S.oy —11
:(j17P4) 0-1111 al*,,1211
0.1..15.111 t 11. 114 J L 4,i.:61.11 „It:LA.11 (33.--11 M( t) = E(e' ) i3) 4.1t1•LI.5113,9 L.4.911--, c.3.1-11 t=0
= M (t) = E(Xe )
: ,A114.3
dt2 M (t) = ex (t) = E(X 2 e x' )
M (0) = E(X 2 )= 12'2
M"; (t) = E(X 3 )
M (t) = E(X` e') : .1?21 l'aSt6.5
.t=0 t A .,-\114 Mx (%) _93,31 A.11.11 r _op; 1:11b_9
— 51 —
_1- 3,1,1 215 CA.-511.11,>"11:1
M t 45s11.4 1.1&14.au ZzALICA:s1.3' jzti
• i x, • • u (Xt)k e =
k=0 K!
Xt + X2 t 2 X rtr + + +
1! 2! r!
Xt X X r tr
:.M x (t)= E(e) = E(1+ —p + 22 It2 + + + ) r! t 2 r ,
= 1 + —t E(X) + — E(X`)+ 1! 2! r!
t r , 2
1! 2! r!
M x(t) L.53 —tr _914 it:. (Li ji r!
: 33.1.1.1O49.4.11:04.11u4h ci;:aka (3)
:L61 1.4 A JS 3a Mx(t) i sac. z÷_si
1) M x (0) =1
2) M(0) = E(X)
3) Var(X) = 2x = M. (0) — (M ix (0))2
: jA)411
Var(X) = E(X 2 ) — (E(X))2 M ix (0) = E(X)
M; (t) = E(X 2 e xt ) = M(0)= E(X 2 )
...Var(X) = M; (0) — (M; (0))2
• Yil•LL'A .316.3
4) MP (0) = E(X r ) = prr
5) M X + b ( t ) = eb I M x (t) b
- 52 -
-1- 31-.:1 1 (..)-=.- 1 215 s.)C-1.
:(•il.k)41
M X+b E= (e(x+b)t ) =
=_ E(ext .e bt = e bt E(e xt = e bt m x (t)
6) Max (t) = M x (at)
:utrzi>41
M (t) = E(eaxl ) = E(e x ) = M x (at)
.1?3 (6) 3 (5) .LJj.1.a
7) M +1, (I) = eh' M x (at)
1;.1**1.21.4q1)SprAC.1121 (4)
c.1_)?.)
L5:11 3_3.1113 ac-1_5.ill
y(x)= Y1(x) 13) (1 d dx
Y2Y'—YY' I I 2 =
Y22 Y2 ( X )
:(Gamma Function) 1.4\.lb (2
L4 `--9)6.9 r(n) 3-91) L5A
F(n) = x" -) x dx 0
:t 94 n LJI313
F(n + 1) = nF (n)
i.e: F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3!= 6,
- 53 -
„.4
13A3
Ca.
Jf'edx= r(a) b° 0
:15
:(Beta Function) 1:1,_1!. 'a.) (3
B(a,b) 141 :)..j,1
215 ;;i1:-. ca+?-1
,311.111 ( c —1 L111431 :k.1L j
,b) xa-1 0 x)b dx= r(a)r(b) 0 F (a + b)
:1.1.4' i
i) x3 (1— x)2 dx = B(4,3) = F(4).F(3) = —3!2! =•••••-•1 0 r(7) 6! 60
a xr dx = a.r(r +1) r!
:(13-2) LA4
A41.1.71 426 :11y1c.y*14 X
, x=0,1,2,....
o.w f (x) =
3X e3 x!
0
.1.L.41.11 X t....5.11s49.,,11 k11.111(1)
. Y j,r;1-11 ci411.3 .-3._5:111 (3) . Y=2-3X Y (.3.1 4A1 k.11.111(2)
- 54 -
-1-
a r+I ar 0
-1- 215 •;_)C-1-
L
J
L:
E
: :1111c. 6.4?.3..4 Ai x (t) cj L.)2mji3 :
1) M x (t) = E(e x
= E f(x) =Z
= e- (3e1 = e -3 .e =
x=0 x!
M (t) e3(ei -I)
-.• M ix (0) = E(X) Mix (t)= exer-1).3e i
...A/Px (0)=1x3x1=3
E(X)= 3
% .M (t)dt
(MX (t))
_d [exe,....0 .3e, [3e3(e, ,
dt dt 3e3(ei-D+i oet +1)
:.M;(0)=3x1x(3+1)=12
Var(X)= o-2x = M;(0)— (Mix (0)) 2
=12 — 32 = 3
...Var(X)= 3
E(X)= o-2x = 3
2) M y (I) = E(e Y')
= E(e 23 )
= E(e 2I .e x(-3 ) ) = e 21 E(e x (-31) ) = e2/ u (-q i ) = e2I .e3 '-I)
1" X k
M y (t) e3 31 -1)±2i
- 55 -
-1- (..). 1 215 s_)-c3-4 _94 C.A1..5111.)-=11-1 .1=-4
(1) siall tas (t) :dam y J i4l i. 3
: 41.1 tk,-.111
E(Y)= E(2 — 3X) = 2 — 3E(X) = 2— 3(3) = —7
V(Y)=V(2-3X)=V(-3X)= (-3)2 V(X) = 9(3) = 27 V (Y) = 27 .
f(x) =r 0
:(14-2) JVIA
A41, *1_4.141 :4_1L X
, 2> 0: const. , 0.W
.f(x) Jj... X
-MX(t)1.,* 11.4 (.1-j171 ZI:a3 ti_gz. X ca_9323 tai.x.11
:f(x) 3L 14 (1 :(.1.%11
p.= E(Xr)
= fxr f (x)dx
= xr .2e -Ax clx =
(r +1) Ar+1
r! =
r - clx
y = Ax 0.5.311r1.1&.1.d1c.A.4.1j
- 56 -
' • 4_14 • Li, _9
1 , 2 , 6 Pi = EV) = , P2 = E(X 2 ) = , P3 - A
2 2 (1 Var(X) = = - A2
:Mx(t) z,-"1 ,,41- (2
Co CO
. A4 x (t)= ex t f (x)clx = J xI2edx
— CO
- e- -x(2-0 =[ i -
- t -0 - t = 2J e-x(2-odx =
0
F-3
r! M (;) (t) = —(1- --)
Ar A
1 215
2 2! liar (X) = --2-A -
1
22
- 57 -
-1- 3-4:1- 1
215 C..)14)11i.r-=L; •L'a-A
( t MX(t)=(1-41= 1+ t +( + + - +
.2
-4j<;-11 Evb Lsi J..
(t r = t r = t r . r! 2 X r! A:
L. 3143
:L53. •kait'SJL (.52113,1c.' ..)4:=:14. X :(15-2) sAA
Ii f(x) = 0
,0 < x <1 0.W
• gr 142" 41.1_9:3
x (t) = E (e x ) = ext ldx =[-L- t
:324
M x (t) = —1 [e1 —1]
M "= [
X•ti 1 1-1-1-4 t1+ -F-tr
+.... -1
t 1! 2! r!
tr-1 tr + +
2! r! (r+1)!
c.,1 ,14_,‹;,11 L.,L9
t r t r r! r! Pr = (r +1)! = r! • (r +1)! (r+1)!
1
r + 1
- 58 -
Li
215 „541 cit? j.g11.1 • 4
:(16-2)
:L i tn5:11' I th L...;3.11 X j,.1:,.:1.11 L.,VPkjl 4JJA1I c.,33.11
X 0 1 2 3
f(x) 0.2 0.2 0.4 0.2
. 4.0_9303 sali...11 :ULM
: I (19.1.11 (L_• u LI) :J.
X 0 1 2 3
f(x) 0.2 0.2 0.4 0.2 1
xf(x) 0 0.2 0.8 0.6 1.6= /2;
x2f(x) 0 0.2 1.6 1.8 3.6= /4
x3f(x) 0 0.2 3.2 5.4 8.8=/1
x41(x) 0 0.2 6.4 16.2 22.8= pi,
P
:= E((X — /4) 1 ) = p — ,u = 0
( D:41 ) cp3L1 L.5.3)-31 c-3.JJ
E((X 11)2 ) = (-5-2
= E(X 2 ) - (E(X))2 = - it/ 2
= 3.6 — (1.6)2 = 1.04
- 59 -
(:)171,..5.9.3H J.4.41-1 .1.4.z..4 4
123
11 —3144+2 = 8.3 —3(1.6).(3.6)+ 2(1.6)3 =-0.788
114 = E((X I1)4 ) = —4u + 6112 14 — 3114 =22.8-4(1.6)(8.8)+6(1.6)2 (3.6)-3(1.6)4 =2.1152
cs-t4 s .4.11 :043
M x(t) = Rex' ) = Zexl f (x)
= e° (0.2)+ et (0.2) + e2' (0.4) + e31 (0.2)
mx (0 = 0.2+0.2e' +0.4e2' + 0.2e3'
Li
Li
X 1 2 3 4
f(x) 1/4 1/2 1/8 2/16
:(17-2) Jti
F(x) L.5-4Sballt-3.1i111;k31- L-3-1)2L1 JS31 0)
Lci LAS1..11.1.1 4 j t a X cjIS
,L3-.111.31 L13-1?-11
Li
.F(x) t...13;111JLLrit.., 2 .A411.....1..ylv-,<I1 c.s.4 f(x) cj1 Etill (1)
. P(1< X 4), P(X > 2), Mx (t)
:
F (x) = P(X x) , x E R : F (x) (1
0 F(x) 1 F (co) = 1 j F (—co) = 0 .1
- 60 -
f77
Cjt:i..)111.).4.=1.1
:D (I.L.1.5-1) F(x) .2
xl < x2 F(X i ) < F(X2)
L'yz F(x) .3
:(2) .4-e31.31 iy94:1_9
:I D13 {X .X1 .X2 X1 < X2 DIS
...P(X P(X X2 ) cy,
.'.F(x1) F(x2)
:a4..),111 aC1:6 (1 (y
.f (x) O, V x 1.f (x) = 1
: j_9-141 01/1
4+8+2+2_16_ 1
16 16
.X 141.ca- ki f(x)
F(x) = f (k) : (:),. :ULM —1 (2 k Icy
F(1) = j (1) = , F(2) = f (1) + [ (2) =
+ [ (3) = 3/4 + y8 =
+ [ (3) + [ (4) = 7/8 + 2/6 = % =1
: jrb
- 61 -
F(3) = [ (1) [ (2)
F(4) = f(1)+ J (2)
C.,%4?..4 CjLi I j.waU 4
F(x) =
0, x<1
y 1<x<2 4 ' < x < 3 4 ' -
y 3<x<4 8 ' 1, 4 x
3 P(X > 2) = 1 — P(X 2) = 1 — F(2) = 1 — —4 = 1
1 3 P(1 < X 4) = F(4) — F(1) = 1 — —4 = —4
mx = exi f (x) 'ix
• M x (t) =(1/ 4)e l + (1/ 2)e2' + (1/ 8)e + (2 /16)e4' .
:(1 8-2)J11
:44 LaS f(y) 3L 4j y J Ls.132,a J4ii4
f (y) = C1 + C 2 y 2 , 0<y<1
• L=4.311 Ci,c2 4
.c2 3 C1 &III:. L;;,!‘ L.4,,L1 2 .f(y) ;Ll:1S. iL L jS31 (1)
P(0.4 < Y 14.1.3 F(y),ASI.5111 .3
.<0.8)
f (y) 0 , ii) f (y)dy =1 f(y) (1
0
:ag:1-11 L,J1_,_‘,..1 (2 I .7
215
(3
:4.5.1
- 62 -
-1- LIL4:1=J
f (y)dy = (c + c2y2 )dy = c dy + c2 y2 dy =1
1 ci [y101 +C2 [L-] =1 - C2 = 1 (1)
3 3
3 E(Y) = yf (y)dy = — = y(ci + c2y2)dy
4 0
c1 J ydy + c2 i y3 dy = —3 J0 4
c [Y2
2 11 +C 2[]1 = 3 4 —4 0
3 c, c2
4 2 4 2c1 +c2 =3 (2)
FT1 :
2 2c1 -F -3 c2 = 2
2c1 + c2 =3
—1
c 2 = 1 c2 = 3 c1 =O 3
{3y2 , 0 <y<1 (Y) =
0 , 0.w f(y) :;61.)
: t.1);1.11 :01.) (3
t 3 F (y) = P(Y y) = f (t)di = f3t 2 di = 3[=-1
0 3 0
0, y< , 0__y<1
, y 1
1-7.1
Li
Li
: a-0141 143.103
P(0.4 < Y < 0.8) = F(0.8) — F(0.4) = (0.8)3 — (0.4)3
= 0.512 — 0.064 = 0.448
- 63 -
-1-
Ls-1111 LIAa=11 LJ&:I.J (7-2)
ZI4c. k3-4 LD:1 J.79 4 L-5_ 1
Lci x L.540-11 LDIS 14i
(c.)
jJ (.19.)4 (y)jL L (1) : ;411:111
JS1 &tall Z-1_)12u . cjtiS.11
? = E ( X ) LrbAl 4-1-4_, X j4iLa..,,L cr,i'LZ141 t-11.9:11 (1
141..3 F(X) :;+.11. (2
?JP 4.51c. s.1_,I3 12).0 „lc. A-14.-1, I 45.1.:3J (3
?,u = E(X) Mx(t) JJ2.11S41 -.1.11.111 (4
H aykla ,3 X j.!;1,11 LDIS 131.iL iJi J 4JA 41.4. j .2
:(D-4 431 PM} = Y5
M X (t), aX 2 Aix L,..L.L.1 f(x) CT x ,
P(X >3) j P(X 4) i4, F(x) (y
Y= 3X-2eLyz.., Var(Y) j E(Y) j M y(t) (E
Y *1-41,71 2-4CI My(t)
?P({T}) P({H}) 141 (2) LI...1cl .3
Li
9
- 64 -
1(..31-Aa... 215 SJC-1-4 C.J1.4 J.,..01..A .1.1szat .J
&al x L;t_c 1-.1) .4
3 r ir , x=1,2,3, 4 L4)
0 , o.w
. M x (t), V (X), E(X) (1)
Y=X+2 L. M y (0, V (Y), E(Y) (y)
411...:141 j 1 X DIS 1.3j -5
77) Li
f (x)
Li f (x) = 13
1 , x=1,2,3
0 o.w
. Y=2X-1 :c11 p. M y (t), V (Y), E(Y) J-C -14
&I... X C.g 13) .6
.1(x) =
, x = 0,1,2,3
0.W
My(t) , E(Y) . E(X) i Mx(t) , E(X) .1 .31
Y=3X-2 :L-11
14i 'DJ A ut:151.4 0.:911 .7
csIc. cji)4,1.1;11 jut! X J.,!;.;^11
E(X) , f(x)
- 65 -
(..)-=-1 215 c: 17J121.)L =-3-a-.3
(31c. j.a:J_1;11 y (y
My(t) , E(Y) , f(y)
z_sic.1 J:111 Lc.") .2.11 (c
= {(a,b): a b,a =1,2,...,6 b
M(t) f(Z) •
(a, b) apax.11 j451 j..!;1,,11 u1.5j
?Var(Z), E(Z) L11+143
:k11-111 (DT 8
-1-
f (x)= , x=0,1,2,3,4
P(X > 2) X j:.;1,11 :;y1, 1.4az. :k_1:i5 LU
X ..11;'-‘1 UtS 131;:s LaY-La grill .9
?Var(X) E(X) 14°3 MX(t) j f(x) 1.01,--11
x 13) .10
Y4.1 4saz..y1 ji 4.31 (2) 13A tj:3 u (1)
TP(2<x. 4) P(X >3) 11411:111 a?„91 (3)
- 66 -
Yy=x2 :Cji E(Y) 3 f(y)
taii.111
x < 0 0 0.13 0 x<1 0.27 1 x< 2
F(x)= 0.53 2 x <3
0.84 3 < x < 4 0.92 x < 5
1 5 x
r71
Li
r-71
r-7-d
-1- 215 ap..J _971 Cil,u11.11
Ty. (x-1)2 p. E(Y) (5) ?X .J4-....1.11* (D,11413 .1?.ji (4)
: 1-1,0,11 LEA 4.4,15 tAg_LLc. . 1}.1.....3.0 X utS V.)) .11
TE(X2) ..1?„)ti P(X = 0) =- 1/2 L (1
Tp(x = —1) j P(X =1) 4._91-i E(X) 16 3 P(X = 0) = ul.S 13) (2
:Le% '4.:15-11S kW.) 141 X ct2)11 c_- 615 13) .12
o<x <10 '
Y= 0.003X2 V11..1.11 y
y Lpt,„..11 Likb.sibs5,11,31 Lw.S-1.4' (1
TVar(X) E(X) (3 ?F(x) (...,-0S.1511 I t.J. .3;11 (2
:Lit .13
.(g czioj b 41) (.341-11 4:111 2.11 (1
X utS 13) (2
i)f(x) ii)F(x) iii)E(X) iv)Var(X)
klf31.,11 Lci 31.ibt)11 ju,.± Y LDLC (3
i)f(y) ii)F(y) iii)E(Y) iv)Var(Y)
— (SLcy. Z c_iLC 13) (4
i)f(z) ii)F(z) iii)E(Z) iv)Var(Z)
- 67 -
-1-J -.:L... 1
01%
10 43.11 N:1.11 40 4:31 t`J1-.1-1 ,1 .14
.1)54_4 j--11 &kVy1L=.1\d/LIJ
-,113 4.iti 0.6 .5,4 csii)S4.4
4L1 X LA,L-tc., LI)13,1L L4LC.. LILL 14 JL*La:171 131 .15
F (x)=
f(x) (3) ?F(x) e...)) (2) q_14;1.11 11% IA (1)
:Z.411:111 L,J.,.) (5) TVar(X) E(X) (4)
(i) P(X > 3), (ii) P(X < 3), (iii) P(1 < X < 3)
tai.111 cj.4 J.S ts.11,y1x11 jcil .16
F (x) =
F (y) =
,0 < x
11—C2x ,0 15_ x
0 ,y<0
YX•
yx 'XY<Y2
y 2y2 <y<1
1 y
F(l)=
F(z)
0
0.20
0.35
0.75
1
0
8 3z +5
,/<0
,0._/<1
,1./<2
,2_/<3
,3
,z<-2
'-1<z<0
,0 < z <1
z 8
1
- 68 -
..3
Var(.) , E(.) , f(.)
4.11 c:J.131 F(x) Li.:"9:111;k11. c)..1...)S3i () .17
:L611:111 csa_.-.14 1 tivil 1 x 4:3132,•ic. (y)
x 0 1 2 3 4
f(x) X 'A X /6 1/4
?F(x) 4,31(1)
P(0 < X <3), P(X 0), P(2 X <4) :ui\ L (2)
c:M(t) L,1-3,1 (4) 4 F(3) — F(1) ti '.6C1 (3)
X 7)4y lul.) M(t) ca-As () :Ulan L._:!) () .18
U
i) E (X) = M (0) , 0-2x = M', ( 0) {M ( 0)} 2
:-42.43ti
is\ Lls N3211 Sa1.5.411 .11.111 .1.31 .19
J3:131
(i)f(x) = 2e-2x ,x 0
(ii)./.(y)=Ae-Au-c ) (iii)f (z) = a ,2 < z< 3
. YF(z) a (iii)
- 69 -
I
16.;.4
-1-J' 215 J 1 djjU ..1
f (x) =__ irlq X-1 , x=1,2,3, :k113 J-..1;6.,. X c_J\--C .20
p+q=1 o p 5_1 j ca?ti p
kn.) LTA f(x) j (DA a_clall
i)F(x), ii)E (X), iii)cr x , v)M x (t) : Ls (Lr,
.A.1‘).41 II% ;AO Lraill (SLA,I. X L:it •sjA
rt)3,012 alc• I 3 a.1_,11 j_541. sic. 1 ‘.,-;<1 a4,4 „:4
.E(X) (c .f(x) Z.41 lk.L_C- 11 "A.11.1 (y
:LJ 4.11 .22
f(x)
: LIS 4TL'A.- 1
(i) E(X) (ii) F(1.5) (iii) P(0.5 < X < 1.5)
- 70 -
J.; C. j174. 215
.A.11.1.11 St..4.414 „4:1.4* X LA4.1 .23
0 x <-1 x +1 12
—x
+ b
2 < x < 5 4 1
5 <x
LI (i) b , (ii) .f (x) , (iii) P(X > 0)
C?1 W,-13 .L11,9.411 WI L51,44 .3114 M X (t) = L—E(Xr ) c-144.1 Q1 .24 t r
r!
M y (t) = r>
:44.1th Y NioLl 444 9 x
.Y j Lflaka II j-9 421.41_9E(Y r ) ds2,21
b*:k_ZI 21 3 20 aa-AJ ir (1) ....4111zNA
1+1)+1)2 +1)3 + , b <1 • ...k+,A) 1:1.4.11 Dt_c 13)
(i) Eb.. = 1
k=0 u l<=0
r (ii) 1— b" Il
1—h
= b)
(iii) kh" = (') k 2 b)2 k,1 (1 1))3
n(n +1) 1+ 2 + 3 + • • • + n =
2 A1C- ((,-3)
F (x)
x <2
- 71 -
-1- L.31--11 215 si-Cia
r°1
LøAoJI IL 11 calsui9J1 •
( Discrete Probability Distributions)
;k 21...111, .111 SJ,L'S c..1*l. JJL1,-411:14
71411-,c -A-616 all bae0 daZT:33 LyAL:1 L5A L5a^11
sakya :61-1113 t-1.3;111 (.)-.1.3& 0:=12-3-9 F(x)‘.1.1.49 2JL L5111
:c5113. C-.16U.141 Oat0
/4-43.35 (1-3)
(Bernoulli Distribution)
-;1.A.114.411 (1)
c.:11.1111.1%) "F" kI 3....):13 Failure SL "S" 43 Success
I.* fS , F} 344 (44.)).1.). 4.41c3 kf_1_52a.
41 {1,0} ccol X c141 C16.4 j..1.44 X 1,03,1c." 6;t1.4. latt, Lcla Lija
• X : {S,F} ---> {1,0}
41 (1- p) _5A p _5A c1÷111 J1_4:1_,* 1 ui 131
P(X =1)=p , P(X = 0)=1— p q , p+q=1
- 72 -
.f (x) ={:x 0 , o.w
r p x .9, I- x , x = 0,1
11)
,.!ji(:)171 J.1.11.).4.41]
• • 13 41G • Li,
: L54 X"A4.11- "*.diS)1 :k.11. (2)
f (x)=
p, x=1
1—p, x=0
0, o.w
1:14 ççA p
:thS f(x) c:_LO • LD.1..),L
LraS1_5111 :;illa (3)
I
0 , x < 0
F(x)= 1— p , 0 _. . x <1
1 , 1 < x
(4)
p = E(X) = xf (x) = Ex jf qIX
x.. x=0
-=o+ixixpx1=p
3z\--) 3A1 X Cji 1a4 p = E(X) = p
Var(X) = o = pa
eya9:111 134 a4143 (5)
- 73 -
-1— c_31_41.1
•LisL4 94i CJI7L):111_Y =11:' .14--4 •
:cjIAJ..111
I ( 112 = E(X 2 ) = EX 2
x.0 \X ..*o x2 = E(X 2 ) — (E(X)) 2 = p — p 2 = p (1 — p)= pq
M x (t) = q + pet
c‘43.53.11 :k_11_111 (6)
: J411
( 1 \ M x (t) = ge x ) = E ex(
x=0 = lx 1 x lx q+ et xlxpx1=q+ pe`
:(_63.URS _.932111'.21•14.11 4 i4 LSYJc--131-11 Ai-1.1 47,11 Ci-C4-19
• M x (t) q + pe t
▪ M ix (t) = pe t = p = E (X) = M. (0) = p
MI(t)= pe t => 14= E(X 2 )= Nil(0) = p
a-2x = M (o) — (mix (0)) 2 = p p2 = pq
p', = E(Xr)= p cal-A
: (1 -1-3) 3I/L4
t-5113j 2 L,Ic.4.A. j 8 L.531 0 D.4 jti:t (1)1 (14
• _,4% X CAI.1
:L:}4 -1 .91 (1)
(i) f (x), (ii) F(x), (iii) E(X), (v) a 2x , (vi) M x (t)
:cjA LIS4.jti ,calt;i a 0 b Y= b + aX j_CI3)(2)
(i) E(Y), (ii) cr ) , (iii) M 1 (t) , (v) f (y)
- 74 -
pt
-1-L)-.1..1 215 1,...)-C:14 CJL.4,59.31 .143-NA
::‘,1)1:111 c4.11 (LI L,:i1411 cb 2 L.53z. .11,L.11 j:Li&.)
X(0) = 0, X(1) = 1, X(2) = 0, X(3) = 1, X(4) = 0, X (5) = 1 , X (6) = 0 , X (7) = 1 , X (8) = 0
: 4,11,c-3 {OM 4-A LA) X
5 4 f (0) = P(X = 0) = , f(1) = P(X =1) =
X 0 1
}, 4.14 c.5.1c. 3A c t.-111 13) 3
4 P(X =1) = p
(1) 4c.
( i ) f ( x) = v 9 ) )1-x
, x = 0,1. x i
0, x < 0
(ii) F(x) 59 , 0 x < 1
1 , 1 < x
(iii) E(X)= p = 4
(v)Var(X)= pq = —4 5 20
9 9 81
(vi) Mx (t) = q + pe1 =-5 + 9 9
Lj
9
Li - 75 -
: 14 (2)
(i) E(Y) = E(b + aX) = b + aE(X) = b + —4 a 9
(ii)o = Var(Y) =Var(b + aX) = a2V ar (X) = a2 x 20 20a2
81 81 (iii) M y (t) = E(e t ) =
= ebt M x (at) = ebl (%+ Seat )
M y (t) = —5
ebt + 9 9
A -41.LJ U Ls.-111 Y J.!'" e-13-11 4,11
: LskII1S C4i.1 _)1.411 JI.A..1.. %\ ,11
y b a+b
ilY) % %
. Y=2+3X (2) ball ac.1:&D.4
(v)
ci
L-3
(c.g=lt-.121 CiTlaa-11 0:1 N,3.3.-.1 (2-3)
(Binomial Distribution)
_lac.) X J.2;1,11 t_):13:331 LiA
LL11).21 (:),A n L54_93 JIA1
ia,L* p c1.41 Ljtg .(csi&T Lsic. :443ts.o ;1÷2.,11 j1Si'V
:
T-1
- 76 -
p q 11 - , x = 0,1,2,....n "XI
0 o.w f(x)=
,u = E(X)= np
Var(X)= o- x2 = npq
1.).441:1 4
fl
4,6 Z3ls.11 2k..11.! (2)
. b(x;n,p) 1314.1 kl . p_9 n taoll 13th
:LA CrA111•21 4'171.10 ;k-11.1 (3)
F(x)=I , r = 0,1,2,3,
c.5 ‘-') JJ
75.j.i111 1:1111 ta.3.-1. (4)
7.1.19*-1.11 1.24 C.):1.LP (5)
= Vnpq :.5t6 L,u .142-4 "4s'41 LD1-i 42",9
> : C cci 431 .);..v ..;k11-%%t-4
mx (t) = (q + pe ( )" :LA -,).19:111 j‘32..1 i>11,5.4.11 2J1.1.11 (6)
(7)
t4 t_333:1' X DIS 13) :(1 - 2 -3) citLA
f(x) c4i (1)
= E(X)= np Var(X)=o- ,2v =npq f(x)
Mx (t) = (q + pe' )" : crcra
At= E(X)= np , Var(X)= a =npq cLL,6
- 77 -
x ( n\ p r q n-r
U.
LJUJL Liu, 31411 1314 Lci
: aV1-.121 (Lrl
n n n"
(a + b)n =
r=0
( n\
( n —1\ =n
r —1/
_ 1 _ L11 L.,--•1215 sisL. CJL,u121.).-=1u 1.,-N-4
:J-
(i) f(x) 0 V x , (ii) f(x) =1 : (:)...1! ).111 3 f(x) L:5-4—n*, (1
: cc:112i .1n_y_tiii LAI c t_1331111JL Liuj2:3 us* f(x) 0 Cji
( n
)pxqn-r = (p + q)n =(1) =1
x=0 yc
k1:a JL L5 L f(x)
,u = E(X) = xf (x) x=o
n rz =Ex pxqn-x
x=o
( n —1) x 17- X — x P q
" ( n = npEpx l qnX
vx —1
x=1 y=0 x=ny=in :( JJ m=n-1 y=x-1
m = npZ pY qm- = np(p + = np xl= np
y=0 \ Yi
- 78 -
-1- L.31-1-1 CJI:459.11
:c.):,.t.32 L:11.1375
• o-2x E(X 2 ) — (E(X)2 )= E(X 2 )— 12 2 p2
Ix2[n ]px qn-x x=0
•
.X 2 =X2 —X-LX=X(X-1)+X
... E(X 2 ) = /{X(X —1) + x x jpxq" -- x X=.0
17 n
=Ex(x-0( jpx q''' +Ex jpxq- x.0 x x=„ x
" (n n n
,, n! =Zx(x 1) px qn-x + np
u = n(n —1)p 2E n x-2 n n-x + np
Jth m=n-2 Y=X-2 -`-••=1.54.5
712! Y np E( X 2 ) = n(n —1) p2 y
Y!(rn p(1
Y)!
= n(n —1) p2 x 1 + np ,2v = n 2 p2 lip 2 ± np _ n2 p2 = np(1 _ p) = npq
• E(X 2 ) =Ex2 f (x)=
n
M(i) = E(e x1 ) >e XI az-0
pX q17- X
\ X I
E (pe t y q„-x _ ( q + pe t y
' .pe dt
M (0) = E(X) = n(q + pet )" xpxl
=nx1xp=np
• - 1.i eulc. • LJ, .. _5
- 79 -
-1- JL.1.. 215 Jd71.5221.›,=u1.-
.11111, (t) = n(q + pe t )"-1 .pe t + pet .n(n —1)(g + pet )"-2 .
...M;(0)=nx1xp+pxn(n-1)x1xp
= np + n(n — 1) p 2
. V ar (X) = 0-2x = M (0) — (M (0)) 2
= np + n 2 p 2 — np 2 — n 2 p 2
= np(1 — p) = npq
:(2-2-3)
%30 j 4 7\1 (:)..1.411 Lsaj
LjLJ j-C-11 0.C2J.44 C.1.4tB 5
.L1c. Lrso.4 LE:11.3.1231
ditil (sic, 4.pj LYL y5l4 (2) .X kta ‘,.Lis• 1(1)
i.gt (3)
.X N_37.3 S.1.13-an U111 (.41.C. 1 (5) .4)1.4.%.4.11 L:21.)41 (4)
J.:;111141'..t.1.11.:141;k115.11:tn (1
(5 HLi
f (x) = )(0.3)x (0.7)5' , x = 0,1,2,3,4,5
r-
P(\.4 LISCA7 L.,-Ic• 1) =1— P(X = 0) (2
= 1 — f (0) =1— (0.7)5 =0.8319
p= E (X) = np = (5)(0.3)=1.5 tis331
°x = npq = V(5)(0.3)(0.7) =1/-1-. =1.025 :c5. (4
M (t) = (0.7 +0.3e( )5 2Jii (5
- 80 -
9
-1- c.31-as- 1 j..31fit:i..501..›Lial 4A-1.A
:(2 — 3) 6.3....)-4:1 (8)
.L1.3 sj4 A:613kLA LA 400 cLia:s. (1)
(H) 451'31 1-6-6-9 cs:111
P 1)
43-3 1.613AU4 X &l..0 131(3)
T c.511.41 km...1)3:1* ki4tP3 kLm...4:310 L,_um.49 X t_idg.1
..J4W 613US (..)-ms) 111 y'.13. %80 r_91-11 OA 61Si:1)(4)
:tã‘Jii.111:26 C.15.u.J-14‘,11). 5 "4:3191c.
JUL.,' 1(3) T&I,111 Lit (2) CI ,g-3.4 cji JUn-,1 (1)
(.J'i{11 „la .1..13 51 (31.1 (4)
Tol-.1AL 51
43 4 i1 (6) 4)14.31 A.,11,11 (5)
:1211-11
<ki,41113)3 .4:1 .1÷:j &A, %20 ,a, (sal A4.4 •ci3IS 13)(5)
10 Lci
4 L:2\611 cds.'/1 L.,1c. LJ J"<1.1,...) J1.-1,1(1)
(4) v_);:,-11L:_ls_91-11c.ci (3) TAI Ls.la CI 6.4
-L.11_9.31 .61..111 (5) Ta1.141:k.,,;.i'i 43i TL:c1.4.11kAA...1c11))..Iti:93-.11.1.1.11
El
1:1
Li
- 81 -
_9,11 L';1-1_}..2.11 jAct1-1
• 5 1 (sic. J-\1 .3A C - .L--g 7 -1J:' (7)
-,1t4 c )..c.2±.; 5 1 ji LsIc. J 1
(.4.6 (3)?Listl -b4 5 1i Lsla 11 L11-.a.. .1 .2
T t._93..11 S.43.411
El .8
• A-44 411. aLi.1 41.
4 ji 3 LsIc LI • 1313 Lsiji 1 Z4141 j1:1&.1 2 ji 1 ,ri_)11 c5ic. 0
:LU J J14 t.;1.41.4 A.V.) tiaL3
. U.1.+1S A.41 LsIc. (2) i_111.41 Ls-lc• (1)
. :1 LsIc (3)
Lija4 L5_11 4).1zi % 2 .‘14 y..1.5-411 &L. 9
4).1311 csi SJ3.?..3.411 - .1.1,3 0- 3 o-2 p 4...31.kli 10000 tsia
5 y ii Lsic L15, 40 L4 ( • ../}i L.5111 .10 Li
Y=n — X j1 X — b(x;n,p) jj ji.11
b(y;n,q) "Li.f31 (1
ç. E(X — E(X)) 2 0 . . .14 (2
Li
Li
- 82 -
(5\ 2 j 2 5-2
cjc ,.9 . S 13L.1 ! 43 Ll..,33-3 041 1...A.5
(X P(X =r) = r—] p r e, X—r
Jl-
4TLILLI1 e71.ii:A. (3-3)
(Negative Binomial Distribution)
::4-4:114 (1)
.1.3j3 u4 t:yalujsj
r 34.3 baija 3.1c. 41 .n cjj r *3*_)
c.:491...,11 larej r 3.1c. 41_9
SA%L3JJJJ n = r, r+1,r+2, t,?1 X _).._)11_ 4.1 n
41. LAS 4-,VA
yia-11 03tc-i (2 — 3) ci-V-z, 43 (3)
sik.3S 1:163.5 Lya.)U1 (.11,11 33c. 'A :k41÷) Lf..1.1t LEAc.
L.53_9 `47 :111-1,1 L.M.&. DA DS*1-+.!-, .)A1-.)L LlL jy
P(X = 2) =
p J L31:3.,11 .6.1.1.5 .k4) cL.
csIPI :;k1)1_,.)
ag331.,11 tjk9 Lyi3L.11 ji),!!\ .11c.
:
P(X =3) = (5
p 2(15-2 .p 2,
3 5-3 q
- 83 -
-1- J....L..- I 0.=-12151;fia... asz.
:111 LrIc. J,--
diriS Z114) Ai) js..:1 (2)
:LsibLJu1I a.1.1.31 t_)js,1-
—1
f(x)= —1
r x-r , x=r,r +1,r +2,....
0 , o.w
.t_i_jja.11 L.A r p
Px =E(X)=.1--- jto 443:1 (3)
41.11-.1:1 (4)
(5)
(6)
2 rq cry — 2
.4.11.111 Mx(t)= i_ qet J
j:0-14 :S4A11 4:.3•11.N.1.11
: o ads (.511 .3.191111 144 ;61.14 Ciy41),1113 C111-LL=11 csi
1 = (1— q) .'
0 —
=l+rq+ r(r+1) q2+ 2!
=E[x qx_r
x.r r —1
:ALA" (7)
:(1 -3-3) Lr1-14
- 84 -
E(X) = xf
r
x(
(x)
r -1 ) p r =zr
x=r
/
p q = rp \ x=r \
-1- JLAII..1 laz„.4
"eilb c2:431 (1
Aix E(X) --r-- c:43i (2
? = 3,54 CZJ-.:3i (3
et Lcro L4,411 ul al (4
q 1- e
1
Li
f) -L!113_, ryl.L.0:141 Lsi.J.Z1 66:1 f(x) (1)
(i) f(x) 0
- ( - .i. (ii) I f( x ) = y prq x_,. = p r v .x
'r -1 L-1 r -1 x-, ,,-,\„ i x=r \ i
p =pr .
(1-
1
0' =
p' =1
0 • L7-331-411
Ii 1.4'37 I (2)
= rp CO (y 1 \
qYi , x=y-1, r=r1 -1 y"
rp rp = rpr(1- q) = r1 = r+1
P P
• ' 3_,\L-431 .Y%.9
- 85 -
a 2x =
• p r e -.• M x (t) = (1 - gel ) r
gel )r.pr .ren - pre' .r(1- gel )1* -1 .(-qe` (1- qer ) 2r
re' pr (1- ge t )1-1 [1- qe` - (-get )]
r 2 pe t + rpre .qe g (1- ge t )r+2
?P r = rP r
(1— q) r+1 P r+I = —r = E(X)
-1- L.31-41 1 215 siSL.
13.2 s.11_,J (3)
(1- get ) 2r rp r etr rp r etr
(1- qe` ) 2r-r+1 — gel )1'1 (1 _ ge t ) r+1 ?pre _ rp r etr (r +1)(1- qe )r .( —qe1 )
(1- gel ) 2r+2
r 2 pr + rpr q rpr (r + q)
.* M(0) = o r+2 Pr+2
62x = M iji( (0) — (M ix (0))2
• 2 r 2 + pq r 2 rq
• a x 2 2 P 2 P P
r 2 + pq p 2
:Lck e_33.1.1sa13.,,11 (4)
pet p
r e
rt
1—qet (1—qet r
:DlAj411
Li
— 86 —
LiJ
Li
-1-J1...41_,.1 v--1215 j4.11.).4.62U
M x (t) _ E(e xf _ LeXtf(x)
= L e (x — 1\
r —1 Jp
r q x
x=r—
( X (ge t ) X p r cr r
.1"=1"
= P
(gel
o r! = pre (1— ge t =
(1 — ge t )
• Y."1-411
:‘,51,3 t.2331111 cnci
cr 2x = E(X) = qE(X) = pc4
:(2 -3-3)J
_91 Jj Ls Di Lgic. 3 _,& A41.11 -c. j..11'24:1S
CjiA _9At r x 02,5q3
:,L*1:5-. A Zi-i- 31 .).36-1; 316 cl:'11 D)
Li
r-11
(gel (
)x-r.(qe` x —1\ r —1
p r q , .g r e n. v r —1
- 87 -
( x-1\ r —1
• x = 5, r = 2
▪ f (2) = (4) IVY(03 4 4
• f (x) = P (X = r) = p r q X-r
-1-J-4- 1 215 JCL.
= {HHH u
q = 1— p =
L
= 0.1055 r 2 4,-4-)
a2x = P4= 2x Y4 x(4X)2 = 24
4.44)
(4-3)
(Geometric Distribution)
::1-4g4 (1)
L,..53113 r=1 atr..)3 y11.21 cj,;N-.11 _133:1.1
csic. ;Ulu, 0,-,4) JJLcia JJ
.(tr.31 ,6113
Li1S1311 Ci'L4J UK .11 3.43 LJ'Net 1.x.o csal LsiK-11 csi
41.21 4:2 Llta (Di CL4_,6
r=1 Llp -L.5113 j y11.11 (:),a..11 ..J.,33:3 Lyd:, 4.14
141i cs4;111 t.23.3:111
- 88 -
pe t M x (t) =
1- ge t
F (x) =1- gx , x = 1,2,
1 p
-1- 3-..-1.=•• 215 "...)-CL4 aAa.A
4.1b(.4#1.II *A-114 4-ki..":1 (2)
f (x) = P(X = x) = pgx-1 , x = 1,2,...
(.31.4.4 X _}.;1.11,3 tn,ail L5A 0 <p <1 ul
• L?3 J.31 L5-1
.14.4ii .L.11.1411ii4.11 L.):=1,y. (7)
ao
Zgx =l+g+q2 + , q<1 A-o
i) x 1
x=o 1- g
x q n+1
1 - g
1 - g
x=0
CO
6.11-1 = (1-q)2
1+q
x=i (1 - q)3
:Ztidzi (8)
:(1 -4-3) JV-i-4
4liL.41 :;LlaS.11 ;1 /4_11. t65.111 f(x) L.11 '- - (1)
c:P (X>x) 4.91 - (x) =1- q' : L5A tia9:111
- 89 -
-1- L11-.11 215 j Cji.71,)111,..y.aU
:LD1 cLI-.!!!7, 4 M x (t) Pe : LDS c:41 1— ge l (3)
0- 2
B
:
(1 i) f (x) 0 L.4.1).7.a1* 1 cy.
ii) f (x) = pqx-i _
x=i x=1 1 p
= p. = — =1 1—q p
F (x) = P(X x)
= Ef(x1 )
=E pe-1 =pqkI
k=1 k=1
1- q x
= P • =1 q x (-rut I- ) 1—q
• L:4311--
(2
0
r:
M x (t) = E(ex = E e xl f (x)
= ext pq = pe t Ze' qx-1 x=i x=1
1 pe` = pe l I(qe1 ):" = pe'
1— qe` 1— qe r
cr,r2 /1,
(3
- 90 -
-1- Jua,. J:11O17.1j0-11. 1:3-1.-%-a
M (t) (1 — ge l )pe l — pe i .(—qe i ) pe i
ix = =
(1— qe i ) 2 (1— qe i )2
p 1 .. • P x = 11 / fx ( 0 ) = (1 — q) 2 p
(1 — q )2 pe i — pe i .2(1 — qe l ) M„ x (t) =
(1— qe i )4
p 2 .p+ p.2p.q p+2q 1+ q
P2 = P2
o = - (mix (o))2
1 + q 1 q = — p2 p2 p2
:(2 -4-3) c3V—i-4
:;4.11:111 J)3.111 D1S 131
i) . f (x) = 2.(X) x , x = 1,2,....
ii) f(Y)= 2 4.(V3Y -2 , y = 1,2,....
,ux,civ,mx (0, 1t1,0- ,my (I)
cs,2
i) . f (x) = 2.() (1 , x = 1,2,....
1 x-1 , x = 1,2,
Lyi_53 lathj
- 91 -
f (x) P(x; = , x = 0,1,2,...
x!
0 o.w
_1- J-- L. 215 _94 (:)1....).).ni.ict:a ..1
q=
• x = /p
32 3
crx — q 2 - — p 322 4
pet M x (t) =
1— get
,,111-1.31 4;6 (ii)
5-3 (Poisson Distribution)
:ZASiAll (1)
.lac. tii.311-:_.)..11:111,...1313.31..1c. J.1..:1 4:311kA9.1.11L111j.;;I,Ilk.,..bal td9:111 lat6
.1.11 L.,itfrAr
:614. 11:415.!. 1 ;LI1A 4.4".1 (2)
f(x) 4S JU LailS 13) y.4i 4i X J.2;1,11 JU
P (x; 2) j-4.)11-.313C 1 1+1 AJ
late JAZ ,LL41f1 >
- 92 -
2e'
3— e'
LJ
,
= 11,3_31,131 ) l1:14 .41-c3tL 4.k.9
M x = eget -I)
13et
%rib 4..A1322 saljA,11:aill (6)
aq..?A.. J.)1 CJI..0.9.1. J.,..0U 143.2..4 ,J
: 4j i jjiZ11.1 (3)
x F(x)=
k=0 k!
.2 J X. tJ1
p = E(X)=
Var(X)= cr, =
131° :k•-`95: (.5 r‘.-4"4') d_91-1 42'3
.-516 Jj i .1.3*-1 (4)
7.13.35.11 111b al.111:1 (5)
:4.1-11-121 FiiJ a9d43:1 6*)12-11 (7)
n L:N51-,11 .11c. LA L...1.1c. ap...11 43 tiii.11 (tip ii)
LLIJ.)1,_p.11 t_593 p
= np
n> 5 0 / p < 0. 1(j_.4:1 L,113111 idziAa..9 • p= Di
.n —> cx j p —O L.Z1c. c;_1.25L1
:;4421 1:4112:1A.H cja2.:1 (8)
:1-643) L18-4 13t4 --1.,)31,1 I
Li - 93 -
215 Si ...1 at.L;11.1.
1
1) E (X' ) = EIX (X -1) + X)]
2) xr_. 0 x!
3) lim 1+ — = e A' lim(1-1I =e 2 n-"\,. 11 i n-4°'\ n
io x 4) lim1+ -1-: x =1 =>1im(1-- =1 ( n-... n
iim n(n -1) (n - x + 1) 5) =1
n.
:(1 -5-3) J114 .1°:4•12. i (9)
=0- 2, caS (2) 541...a..) kw.) f(x) L:j1 ,:*41 (1)
? a 2 j J &AI M x(t)= e2(e-1) c..41 (3)
&3:1. Ls-11t-2:19.-111 (Mx(t) 3 f(x) c:J.0 (4)
n cLu-439
‘,..53.133 3tAi jj Jc.tr, tijs.11' I Ilk. f(x) (1) :3-31
i) f(x) 0
ii) f(x) = = e 2 E —= e2.e 2 =1 x=o X. ..0 X.
(2)
gx = E (X) =Zx. f (x) x=o
AX=Ex.— .e-2 = C AI U x=o x! x=1 (x - 1)!
00 2x-i y = x - 1
= 2e2 = .12-2 e a • = A = A • (1) y=o Y!
n->m
- 94 -
-1 - JL J215 jS
j JJL3 A4..1
• V ar (X) = o-2x = E(X 2 )— (E(X))2
• E(X 2 ) = E[X (X —1) + X] = E[X (X — 1) + E (X)
= E(X (X —1)) + (2)
ja, 4.3L3 X=1 3 X=0 LAXIC.
E[X (X —1)]= x!
2x-2 /1.2e-A ZX(X 1).
x=2
OD )x-2 = 22e-A y = x — 2
AY y=0 Y!
= 22 e -A.. e A = 22
E(X 2 ) = + (2) c)...
cr 2x = E(X 2 )— (E(X))2 _ 22 4, 22 =
.1?- (3) ,(1) CyA3
:c5.4 I.33R-11 (3)
M x (t) = E(e xt )
=Ee e e x! x!
=e -2 e A(rt -I)
x=2
2 =a,,=2 (3)
Li
Mx(t) i4 0-2
M rx (t)= -I) ..1e1
A", (1) = e"' -1) .2e' + 2e1 .ea( ei -1) .2e1 2(
=2e .e2(e1-1) [I + Ae'
=
/1.6 = E ( X 2 ) = M (0) = + = + 22
- 95 -
-1- JL 215 siCah, • ( *JL,4, JA421.1
2 CY x =
(.5-11 L;AY-111 Lru....›L (4)
:f(x)
r"1
( qn-x \x
n(n —1) (n — x +1)(n — x)! x qn
f (x) = b(x; n, p) =
•P x!(n — x)!
n(n —1) (n — x +1) Ax (1— 24
x! .121 A ' In
n —> co Lii
n(n —1) (n — x +1) /1.' ynr lim f (x) =lim
x! 1 j1,51)
ytx e = p(x,2) „
: Mx(t) sa13.4.11kl iJa&::1.44 :.145 fl
M(t) _5414 (113--.1_942_9 Mb(t) L;112-111
Mb (t) = (q+ pet )11
=[q+ p+ pei pr
= [1 + p(et —1)]n
1.4..fic
- 96 -
Li
a M b (t) lina(1+ A.(el n—xo
b (t) = e l(el -1)
M p
(3) 1:413-1-
:(2 -5-3) J1-14
350 LcIc• 131 (:).4%0.3
13.A, Etzli (:).4
63.13...11 13.4 4.9:1 CjiJ (1)
63". -11 Ls-i Lsia ) (2)
r _53,..31;33_14.11 L7ii.c"" 1 (4) .L:53..11‘..11 .11%3 (3)
•..
JUL') Di 4 '. Ls•13:11-1.. L-2i.9 5-'2 :6.91- L114.3
0.3 n=350 9% \2=11 • ..}43t-i-el i3c. _03 p — — 0.003 'itt, (c17.- \\I 3..43-H) 100
•
.c -141 Lsi2I cUl
x = 0,1,2, L•15-4:1 ti. .33'3 t4.4 X DI3 x!
1316 Al-C.11 1UL .4-?1c,,32 =np = (350).(0.003) =1.05 :DI Lfa...1..
f (x) = (1.05)x e
_i as
, x = 0,1,2,••• x!
Li
• • -kaic • L4 -
- 97 -
-1- JE-3-4 JA.au 4..!
1) (1.05)° I°5 e-1°5 =e- = 0.35 f (0) =
0! 2) P(X 1) = 1- P(X = 0) =1-0.35 9(J.65
3) ,ux =E(X) = = 1,05
4) Mx (t)= el(e1-1) = e105( e`
:(3 -5-3)c.A4
1..&1,)t4c-V3.)1... 24 431..9 Lgi j,-11 otS 13) (1)
. 4.3 e.:14.p.11 c:kA,1tali A.ztadi tikg
. 0.0005 JA )9).4 ZI.,.11.4 JI.Aal (2)1 (.5:i (2)
. bt?-. 10000 1-31-4;-1. 1 Y1-64- 1 Y-1.52.3 C IA.1")
24/ =1 /24 :_9A UI 4piI J..12,4 (1
a2c. J.L.1.) X ui LJ
12 1 P (X = 2) = f (2) = = 0.184
2! 2
. ,ux = 2=1 .1.1.113
n=10000 , p 0.0005, 2=np=5 1411(2
f (x) = (10000
2
52 (0.0005)2 (0 .9995)9998 1--:: — e-5 = 0.0842
2!
:(4 -5-3) C3/14
mx(t)= : c5A X (.5.313-tc. j ie 33211 4.; :61.)
- 98 -
1 215
• P(X=3 ) 1314 t--)j.3.3 itb
:JA21
:2= 4 LA JJ:1 e_932.11 4.3:3it çto LA
x = 0,1,2,... P (X = 3) = f (3) = e4 43 =0.195 x! 3!
:(5-3) a4)1.4. (10)
M x(t) f(4) f(1) = 2f(2) j-S3 t-1,333 tAl. X j\-C I:1) (1)
(4i 0.36..4 L31,,a,.) 2000 4 (2)
43ti 0.001
L3-‘3
•56ti u.! --n" (2 .:c)e3
% 1 L5-4, z.31:631 Ls__›41 Lp3-.0 (:Ji t. (3)
:43ti t-aa.-?-.3i a D. Lc u- 400 jti
j..,.,!11 DjA „.! Lci LA/ Ls1r. Li.=tiLli 4 ..)j..?.j 3-4.s. 1 (1
.11131 (2
Jt.a,) jtb1.4 . 0.005 itz, J : 1.1 ,‘c :Lr-142 uls 13) (4)
';k ,,\c 1000 14,, LsAc. ,L..11.1c 5
&JAIL, jtt,
(5)
u 200 ;1,2,4c. c1 .1000 L$3
A:y.1a 03113 \ Lir+1;.1 U271.)..2q (1:14:1. X
- 99 -
215 -1-
(2) ;LII.) (1
(3
J.2;1.11 0.1.J43 0.6 .ya (ZVI u15131 (6)
-Lic• j144. X Ls:1Yt'll
TJ5'311.L1...1 a.41 (3)
3 34 D.1411 Lsa.1 (si Ci).11 u1c. .lac...1. 5:14 D1.5 13) (7)
cgi ,111.11.p.11 LDA:i. X 4.5L132,..I. j.!;1,,I1 L D..11).ilj
:•1•_51A •
Tia_3411 aLL37.11. jtcam..i (2 ct94.,..{11) j Jt4 (1
¶t4,41J Zmiji yelll J4 J1 C);CIA (.11.41,1 (3
Tier.:21,) Jl4 (4
500 y1:1S ;Lc:33.414,4,gU 300 ,511ae, L:j1.5131 (8)
DiJa& (2 Dl.2.:1-u, Dt1=L (1
Di D:c94 .a1.4. 34 J1a.49 kaa. Li4p (9)
lac, &A:J. X J ,11
44,5,6,0,1,2,3} :4..J.I.t.111X 4 sji;1:1.411c2P...,* 71(1
L-9:1 (3 1-4 (2
o
- 100 -
-1- (.3L.1.1 u- 215 LDL.J.
fl
(10)
;
(o111.3.11 0-4-441 Ni._92. c.r4-u4SjjNi..9*-3.11 (6-3)
(The Hypergeometric Distribution)
.1.LASIL4 (1)
J.,.011,31 M N ul_S 13)
2u4c. l'ar4 c :“-11 N-M c
L51c. 1 J M &. 41 L5Ac. Ji.--11 Di 1.3113) 3
M .)3c. 41)L:J*1.›.. 33c. J1 X j.?;"1,kli . N-M 4
ti,13
L,_k5 \I cji : 03fil ;;LIWI
0 p = —
N
X Di L., \,.; 1.43 .;dii_u1.4 L)3 .3 (..5.11:1-L9
4. 3
• P N ,ks—Aj.)
- 101 -
_1- JA LJ 215
L- J J-414 „
p olls L53 t a3.14 cji: 4)1.111 Q1.21
(c.'13L,1) 1:141f
L.313 i.:12 ,-11 43 tiiis X L531:11143 c_N31,313
1,S ;141 •ta:,41 411n< CSLifzi 4331 Le.,1411
(u4.31 (1)
1.1..1c. C3)=L4 u-C49, {X=X} :C.3.11=11 LsIc• a3,--,-113 (2)
c.5.1taltaj (MIN - M)
(IN P({x f(x)= x AT?
nJ
: c.r,31:111 ,_.11)..,:1 1 L5.1c. 414 3 (3)
:( 1t4 ;t1S11 411 1) 4-ka..":1 (2)
x (A4n.4iql ,41 JjØL:1),Ls
:z4its31
(mx)rn- M -x
x — 0 ?,1,2, min(M, n)( /14./
; 0.W
f(x) =
( Nn)
0
-
- 102 -
M N-M N-n = Var(X)= n. .
N N N -1 ( N -
= npq
OV-1)
-1- Li\-411 215
LI n M 3 N H (x; N , M , n) 1+1 _j..").3,3 .
q LI,t0 p f(x) .t..13511114.1 Lcto
M = Np N - M = Nq
, x = 0 ?,1,2„ min(Np,n)
f (x)
Li
0 o.w
421,3*-1 (3)
= E(X)= = np
:,914 421,114 (4)
:Z4+4 (.:ILL.1.!..A.A (5)
. F (x) = (x) j4E. ,C)5111 ta9:111 ZIL1 :;k; .1.?„0 (1)
• 62 " 13+t M x(t) ZI L111 1.6 C ; .1` (2)
j;-.! (.:111_9) x J A L J :1-'1 Ls-L.; L7 -1-?7".( 3 )
X A.3'1 MiNin
- 103 -
0
swi
13) M:=U.„-;'4 X :1.t.) . M > n Laal.S 13) n
.max( 0, n - N + M) Lcii X . n > M Lc
x
(i) N =10 , M = 5 , n = 4 x = 0,1,2,3,4.
(ii) N = 10, M = 3, n = 4 x 0,1,2,3.
(iii) N =10 , M = 7 , n = 4 x =1,2,3,4.
taii.111 144 261.1,31 ao,. 1)4113 L,3 \:s1-1 (4)
:414
rl
fl
7
(i) =n
an 7 7 (ii)
b a + E i=o n
n < a + b
: c)1:3.11 2 kI;14 tf.A.).3,3 ..11.1G1 b j a
a b b\ a+b 7a+b . . E. xtr-t z xtib-t =E xtr+b-, i=0 =, „i=0 ij =0
7 +
1
xo + (a+b)xi
7 2/
1
x 2 +
2 +
( a+b) 2
tb;x0
x +
(b 1)xi +[b2 jx2
7 a'7 0
i.e
t)
n
b +
n —1
b \ (a +b" =
7 all n
xn J l A2Duida-L5
0 =
7 a + t) n
"a"
7
L3
- 104 -
r,"•77-,
t_)..1 215 7,_)C-1.. ..971 c:.)171-5111.Y.41-1
4.0:'1-121 JJ(.6-"1-14-11 t-?..3S.411 (Ty...**1 (6)
IL,41,4 431.9 = np ce,,41 L 9ti.33:111 11..31L1
,114.1* s1c. n j Sj.}.6 N C 13-c:11.4.11c.3 N
.L;A:L1 tii3:111 a3.t?:i LA JAI. (3.3a a.11-121 LD1 -1,-*-1(%5
ji (.--11 'cut— tt?..)) c13-1-.1 L-11 C.13-C--1 D.At4 (Su, Lci
4-3 ‘c;11:::311 ti.)..9*-111 (DA cp-".6-11 ti.19.331 °34 L-114 Lsi 431
urn= urn nix] N - n
1'1-4' N - 1
x = npq lim 1.-"' 1 - yN
= npq x 1
•cp3311 ii):4-3 913
?"9 44=5=1 cri-..4131 &..10 Lry.):3* L5-127! 1:11b 9
(%5 J L.111 c,,on Lallc LLaa..4) n j N N
(7)
:(1-6-3) JelA
.1.4341 2J1.5.11 U f(x) ;;k4r,! ,L— ciL.:4:31
p = np .314
Z.1(x) =1 ul
• ,f(x). 0 Di f(x) L.J.. :d3J1
-1 -
- 105 -
(MyN-M` n
f (x) = E ( N\ x=0 x=o
1 ( N\ = 1
( INP fl )
\ n )
( AT x.0\,-y ) n- x
nJ
-1- (..31-.1- 215 J41 C.)171.51.1).›-zU
p = E(X) = E xf(x)
n-x i (MIN - M\ = Lx.
xO ` = ( X.
( N N x=0 n-x
) \n
= n-x i N j x=„
tfl
AA 1 vn ( M-1)(N-M) =—• IN),1\ x-1 n-x
Air 1 ( N-1\ nM 1 .N
nrrj \ n -1 N •
nr) n -1)
= np
:(2-6-3) 011'4
:0* iLh LL Ujal Lt11.2:341,sla 14.3c. tlp
.1?•_9 jj cil21/1.1, :45 ciA tj...1.&LI 4 3 1;j13.3.11,...1;,t-,.., Li..12,2 6
J.!41-11 cits 1313 •:.A,1-1:1_,,la cri (.1 99L-a4 6.÷.-• 4 31
E(X) cr2x LLC (2) X j.!;-..11::A...,.11.....141 kli-C.* II :LIU (1)
SIZ.411 y1.11 Z.a-C (:).4 (3)
- 106 -
Li
a P(X = °) = (°)
(6)
\ 3) = (10
\ 3
:P(X =0) itt (3)
6! _ 3!.7! = 1
= 3!.3! 10! 6
-1-
J.4.41.2
cJl u dt.4:1.* 1(4)
: 44.1c-i N=10, M=4, n=3‘23:5...11 :3=11
: -;411.41,41 J2 (1)
f (x) 7 , x=0,1,2,3
(2)
E (X) = N 10
o-2x =n.—.i 1--- t. = 3.—.—.—= — * M MN-n 46 7 56=0 56
N N N —1 10 109 100
:P (X 1) itt) L.:4.41.411 (4)
5 P(X = 1 — P(X <1)=1— f (0) 1 = —6
:(3 -6-3) Cila
. 30 L5L, .4.9a. stpL.4 :;\41:t.,,y+S LL.441.1
c:)A A7 1-1: lc • 36-'..)=i-.3 C-.PL-=" 4 41 454. ul (16_,
L.)---Ji.1 6.3 L;4 14?-1- " Lsic' 131i
3
y__> 1:1111:1k, L?11 —11 A.4.,_,z3 (2
- 107 -
-1- L31-. 1 14"-4 .•
4 „lc. LIy --11 LILA:I..) (3
4..1 X .•. X
: Xfli iS.. N = 30, M = 3, n = 4 LsA
, x = 0,1,2,3
:P(X = 0) JA Y.911'"1ll 1
(3127 ICI) 4
P(X = 0) = f(0) = U
(3 _\ - 13° =O.64 203
4
: P(X Li.:.4)11 cola La :4311--31 11411.2
P(X ?_1) =1- P(X = 0)=1-0.64 = 0.36
.%36 Z.1.311-.11A.,, A1 •
P(X =4)= P(0)=0 : 41( c131..3) 4 a&t.4cji '1 X ..),:;',11 3
:(6-3) cJti.)-4:1 (8)
c'1.3..-4 20 1+14 IS 4. j;kL.A.S J A ,t j11:3. (1)
Itejtij :k41=. (.35 c›. LINJ--, 4
4.1 V13 11:1-11 L1.6 11,.!1-, t4JCPcIN.3-,11
z.„.4 L5l4 Lai cC_I'Vj_v0 C)c. jr.L.JL.4 Li<'tj 0.1
Lci -JI431.).1.3114.3 cLifi.t-11 aziÜALer'--='AJ Ls:131
B
a
- 108 -
-1- 3_4:Ia.' I u-.1 215 ejL.4.51.11
6_93i...1113A t.2)1 J 301+1. 100 & 63.111-al (2)
1..13:111 .L1c. 21-4-j X L..s V:,..11 J.;;n.11 Cil.0 131.i
Le...0;4 11 cio 14. P(x = 2)
U4 UA3 Lplit"-'11
(Jhx:621 tsk, ;Lac. al. J1-41 (7-3)
x uts 131i ,7 LgIc. L:16.4 Sac. ut1151A J 11.3JA.) Lsib (1
csi 7 t9^,11 Lslc- c_13,---11itz...) (1) A.,33U14.1.91133c.
33.11(3) Lcic. tiji 3.4 7 ty:-11,1r. JIA:1) (2) ..;Lel.a.
LI_"A..1.,4_>11 j1.!, (4) 7 ty-,-,11 c5Ac. Ls:1 ;L451 11 C.:442)11
I1aiali) X J.;1.11 eij.3 sal..5.411:01.111 (5) .7 ty,-...,11,1c.
ji 5 4_91,y. Lly-,11L Y _x;1,11 utS 13) :64L21 J1:3,31 Lsi (2
•ZIP:111 A+.. Lci Jill (1) • ti,.11 L.5.1c, 6
si.1 (3) •clit,L..j 6 c.1.)‘-L 'L:43\:. (2)
c.-zitS 13li;k= 14 LIS u --, % 1 0 Ls-th .*;k t 1-4 ( ) (3
j ( i 114)U1 L:Ittx.,a23-,;11 L11.1.4 X J.,!;1‘,11 1.9 L:152, All Ls..1) C111&.31
: y ja3.4.11 c o2kij
•X N32c-i }.343-431 :OLD cict, .2,13(X > 10) 3 P(X = 4) A.4
4 .P(Y = 3) ' My (I) 7e--/ _ 3 :Y J.1;"1,11 7)113,411 :U1-111 4.11 ( )
4-j 1.6111-i-,.. C...)t-5 13) O.42-4 Lris' .4-14:41 U^A• (4
0.5 JL ,C9.14
:th4.41.L.31
4iiL.1.4.11 "L‘).9 (1.9Li LIAO Lry.i. .471. JLA jto La (1)
- 109 -
_1_
J L j1-4.1a,1 jA L4(2)
ay.1 L 9 u(3)
d.IL 4-4141,9 4 jj j4Clall 171,1011 Z.+11.Z.z.* ;US' j1„9.1 411 (5
LA, Lal
e t
(i)Mx(t)= 4-3e' (ii) Mx (t) = - (3e' + 1) 10 (iii) Mx (t) =
i (6
10 (.5-1.c- C.A9 CDA JZI 'D•IA
-A4_,-,L5 43c. L111 Id...11) 4.5j. sic.
EI-L'ul D-4
LJ LJ :A.; c.}.4 2%
•
14‘1
ji.c.10111 L.J1-4.4.CLI1 ( JJ LI-C2q J1.1
Mx (t) = (et + 2)3 X (5:41.3' J.1"1 e_33.3.1 saki-.11 131 - .P(X >1) JL-Aa.
ut-CJ ljJgycl-24 J-1"" L:JA X .‘ 11-3.3?-11 UIS 13) "1 (7
jZti t10-411 (.3 --1 U p(X = 0)=
• 0.9 (3)l.1=) J 2i j 13)
4_51, ;1-,1.11 oltt "bac (31.)11)
(31_4.1 19 j1_,.=11q J.4.11 LiJ 4L (31,1-1 0
ZITO
c1111.3--J5L4.'.041?-‘;=LI J4c:. C)-3 -0.)-4'-; 20 :1-c-JjA `"
*;_j_a L:3 j..11 tii.11,-= J.2111L cy.
J.46,
- 110 -
r-7 ,
r
fl
-1- c_11-41,.." I 0---,•1 215
zotAJ-4
kJ lo.1,>. II 1 u LS&J
(Continuous Probability Distributions)
Continuous Uniform Ditribution : 4.?.6-11111 (1-4)
:LA (ZA. LA:Az.* yl'Aka .A.11.) ki (1)
czi31.5 13) (a, b) 5X1 c ltr6-11 t41.1*.. X
: t Alr is** 'US 61..)
{1
b—a f(x)=
0
a<X <b
0.W
.t6.9:111 t14.1. lA (...hli.2•4, 01,1_1, a < b
Rectangular DistributionJ.:14--11 t....33:111.13.14.1 jii
L5 JLL4 jc,t: J.!;•;.11 N's* tiA9 j_c111 :14,1.
4.5Ly ,1113-C31,5-1111 L15-1, sfii („5:1 c.1;•13 X tii_9 • (a , b)
J1.1c-S1 I dila?. (:).3.5..C:a. L5A 41.41.1&a,A. e.6313 •a9LB c.53jL3G LcJL;s.1
:LA ".444ifill 4 jj .A.11. (2)
F(x)=
0 , x<a
x—a , a_x<b b—a
1 , b<x
a+b = E(X)= 2
etb eta
t(b- a) M x (t)=
-1- 215 siCa.. • ,J
(3)
.41.3.11 med = E(X)= a +b J.-4:1All 431-'49 CD1 2
2 (b- a) 2 Cr X = 12
4.4,9)4 bJAi (5)
-4.11.31 taa.:,:.L.1 4.41 4411 t 'AI1b A LSA .:45•••-.13
:N3231
br+1 -a' = E(X')=
(r +1)(b - a) , r=1,2.....
Y=F(x) ..).!;1,11 „DiL gkai i.asj lath a
oat:,51,....1.* X caLS 13) F(x) JL Lici (0, 1) sji411 (sic- tiain
A+Ityt4 th-1171 c.,1Ll 4, Liu.;
(6)
:(1 -1 -4) Lrl-L4
. F(x) (2) f (x)dx =1 cji c:411 (1) a
= E(X)= a +b 0.2x = (1) - a)2
12 • 2 c431 (3) U.9
.1 4.3.."9.D Mx(t) cim.11 (4)
°
r11 :1
0
Li
- 112 -
F7-74
n—q • q -5x5v`
v—x
n—q n—q n 17
= ip f = f =
(x -5 x)cr = (x)g
= (r)g ••• V - X
0
triuj q E
z)
t'77.1
,v - q - q
=v-q
1 f v—q q 17) q
trvr:
asz.
P xf (x)d,t a
(3
(b2 a 2
1 a b—a 2 1 [
2b—a (b — a)(b + a) b + a
2(b — a) 2 x 2 1
1 [X 3 b 1 (b 3 — a 3 ) E(X 2 ) =
b — a 3 3(b — a) I vi
(b — a)(b2 + ab + a 2 ) = b 2 + ab + a 2 3(b —a) 3
..•o = E(X 2 )—(E(X)) 2
= b 2 +ab+a2 (a+1,) 2 3 2)
4b2 + 4ab + 4a2 — 3a2 — 6ab 3b (b — a) 2
12 12
tJ
b) ;jail ci t4i çS b_a ZA.g- ±31' c_vt4:111 11_44 (Di 13_,',1
:416 N3.11 salj...11 61.111 (4
M x (t) = E(e xt )
1 1 re' ebi — eal
= ext cbc = b — a a b—aL t a t(b — a)
: Z.1. J.3.111
b r X r+1 lb I L 1 —a E(Xr) = j. xrdx =
b — a a b — aLr +1 b —a r +1 a 11 Li
Y ." — a' br+' —a = (r +1)(b — a) (b — a)(r +1)
: (2-1 -4) Jtia
a > 0 , a) Zjall TJ'..""3 1-'70.95 X LDLS 13)
- 114 -
1
11
_1-
215 _94 JA411-1
P(X >1) = i a
:L1_N21
1 1 , -a<x<a f (x) = a - (-a) 2a
1 dx 1 [xl, 1 P(X >1) = 81
2a 2a 3
1 1 —(a1)= - 3a-3=2a a =3 2a 3
f (x) = -1
, -3<x<3 6
:(3-1-4) 314
1 -1< x <1 2
(1)
j
(2)
x E(X r ) 4)(4) •C (,..33T\\ LI, 45.a :=U1.11\ L ,-,c1 (3)
&
b= 1, a= -1 (1)
F(x) = .1" I (t)cli
0 , x < -1 x -(-I) ,
2
1 , 1 < x
- 115 -
x) =
„94s JJU 215
F(x)
LA. a + b -1+1
= = 0 2 2
2 (b - a) 2 0- - (-1)) 2 4 1 ax = =
12 12 12 3
(2
(3
Mx(t)=e(th bla 2t eat
t et —e-t =
r+1 a (r +1)(b - a) 2(r +1)
1-1 0 414 = E (X) = = = = 0
,d2 =E(X 2 ) = -(-) 2 1 == - 6 6 3
(3-2A, = E(X 2 )-(E(X))2 = -0=X
141 j.1 4_,111 411 Ly,i3
Y4 X1 P(0 < x < 1/4) = f f(x)dx = - dx
0 2 0 1 [ 1 xJg.1 1 2 2 4 8
— 116 —
(4
(5
ftj
LJ
(J14 ji.11,),..nU
-Y2 1 r P(-3/4 < x < —1/2) = f(x)dx = -2- Lx]44
1[ 2 31 1 1 1
'cjp."_51i Ls1c.JL &lerio tht * .:L}Lial:. 1 12_.:1L3
.1/4 =_ L.3313.11
:(4-1-4) (.3114
LI-ay 13) .Lk+L-,3 k2,11,..21 kc.U1 ic.141) 1.64L 15
4:11 t kc.L.113 kr.1.31D,L). L.53)
*J-1
L.J.A SAS.) 10 (2) 5 (1)
-1-
L,14, (.34.a•
. 30 u.1) 15 ap. ji 15 ())-11 DA ic.IsL3'.) j11;;',.,71DAj ji X
L543 lAi
.1...L463... 7:30 7:15 :kr-L...31J-11,3p
:LcAi 1c4 a) 4/1 (1
= 0 , bT =15 & a2 =15 , b2 = 30
P(X 5) : JJ (.5.1P1"0:5)ili 3LL
P(X 5) = F(5) = 5 - 0 = 1 15-0 3
: p(x ?_ o) _.914 L._1_,ILL...11 (2
Li
Li
- 117 -
215 _1_ c.J --- 1 ' Cji,J.59.31 a J
P(X ?_10) =1— P(X 10)
a + b
= 1
0+15
10-0 -2- = I 3 3
:E(X) (3
0=1_ 15 —
2 2
:(5-1-4) citi4
.(y2,y2) x uts 131
-1
El
f (x), F (x), M x (t), P(1 X 2)
0 , x < Y2
x—Y2 , I/2 x
1 , X
[-J1
b — a 2
M x (.0 e th _ e th e Mi _
X 2) = F(2) — F(1)
= 1 — (1 — Y2) y2
13U1 ) f (x)dx lito LIU:141 2
[3.
t(b — a)
P(1
— 118 —
Ae-Ax , x 0
0 , 0.w f( x)
11
-1— c.31-4 1
(Exponential Distribution) 4.0-41 Na3:111 (2-4)
:LIA (L ;t.-I14) 4-4J31 (1)
4:gic *a.) c2,3ts 1*.)) x
•t--133.11 _o_9 > 0 Di
Ls 11 L..i 31 t433-311 1:1A,
• J11.1r671 -43) Z.J.1.14.11
Na_9:11. 1 .431.1 (2)
F (x) = P (X x) =1— e-Ax , x 0
1:1
FL3'
LJ
:Lpi . .3..as) t..1k
:43-1ts
> X) J •ULL,'p_5 p(x
10 , x < 0 F(x) =
1— e , 0
..).2,_yj -Ljg U..44.L6 4.13 4 OA
(x)= P(X > x)=
441c. . X u.o J 31 6 -..11
:.314 4.311.3.1 (3)
- 119 -
c..•ILLI.A (7)
(1)
, t <2 : S .11321 A.11 .111 (2) M x (t) = t
P(a X b) = CA° — e-2b
-1- J1 -1-1 L,--1215
2 1
ax = —22
14:i (4)
; JA (5) (5)
, t<2 M x (t) = 2-i'
:(Lack of memory) uall" .4,43Li. (6)
03A Cja j42-1.9 \ t-iig :)...1.41.4 Sig all Cji Yai ...‘46.42t
P(X > x+ ylX > x) = P (X > y)
1 M x (t) = =
1 — 7 < 1 : 1.4.S 14141.1.c &SALL,
:Lji ....t.4142Y :41:314 bt4c j.43
Mx = YAYI = tr r! r! tr.
= _ r=0
= E (XT)= —r!
t r • M X (t) Ls3
r! 346-4 Ya LDI
L4 i1 Sa' t..13.3:111 14,1 aL,r4I_A j (3)
0 I Xa-1 e-bx dX = r (a) (l431Z ba 0
ii) UdV = UV - VdU
: 414
- 120 -
-1- (.1k- (.)--,A 215 •;.).L-A
:11-11,4i (8)
:(1-2-4) dt14
rf..73 LL),i31 LrACII Lci
00
F(x) = e (3), F(x)= 1- CR' , x 0 (2) f (x)= dx =1 (1) 0
• ( Mx(t) f(x) , cr2 = x 22 (4)
F(b)- F (a) = e - e-Ab (7) dack of memory (6) ,Mx (t)= 2(2-ty"' (5)
:(.1z-11
- e-axi°D _ 1) .1. f(x)dx = cbc = = —(0 -1) =1
2) F(x) = P(X x) = J J(i)dt = C A' dt 0
L.D = [e = - 1.1= 1 -
3) (x) = P(X > x) =1- P(X x) =1 - F(x) =1- (1 - C A') =
:f(x) Lii4. 0-2x _9 E(X) :%1 (4
= E(X) =f xf (x)dx = xe-A' dx = r(2) =! 22 0
2 1 1 2
E(X 2 ) = x 2e'dx = 2. 1-(3) = 2 0-2x = E(X 2 )- (E(X))- T2-- — _
2 - 22 3 22 0
:Mx(t) 0-; J E(X)
A M x (t) =
2 -t =
M'( t) = A(-1)(2 - t) 2 (-1) = A(A -t) 2
E(X) = 114 '1, (0) = A .2- 2 = -1= 1
F-1
Li
- 121 -
-1-
(t) = 2. — 2(2 — — 1 = 22(2 —t) 3
11/17v (0) = 2/12-3
62x A / f (0) — ix (0))2
. 2 2 1 1 • • ux 22 22 22
03
5) Mx (t) = E(e')=eXt 2edr
F(1)
/1.1 e-'(A-I)dx = = = 2(2 - t)-1 (2- 0' - t
6) lack of memory (*)
P(X>x+yIX>x)=P(X>y)
L.H.S=P(X>x+y1X>x)=P({X>x+yln{X>xl)
P(X > x)
e-Ax .e -2y P(X > x + y) e-2(x+y)
P(X > x) et' = AY = P(X > y) = R.H .S
7) -.• P(a X b) = F(b)— F (a)
F(b)— F (a) = (1— e-Ab )— (1— e-Aa) —26
=1— Cilb — 1+e' =e — e .
: toa.c Lei/ Üli J L (*)
44 Lsi&cp., 411
J 'tjfl jji&1.1'.. X L.,ilytc. JL.5.1.11- I tru...c. t:JIS 131:1 (x
JUL,' L5,31.,a), x saAlL.ILc y sa..1 ijh:i
.y O.1,11 -1).1...1421 -AST
- 122 -
215 k.)-C3-.
:(2-2-4) c3V-L4
t41*- :;y.)'lDe,S11 Ly...31-11Li L13)
13.4 ;41.3.Lac• ;Cu" ji..4 .A.a.L. 1500
.A.c•L 3000 (:).. ) (1)
.Ji cyA 150 (Pi& 1,—.11 C5.51.7,3. L:)1 J (2)
300 b. A.-4 (s)2,1 1200 cl.,---1111% L.11-) (3)
_
E o C.).4 j1.4. X ...).!;*.t-,11 (Di 0:clii4 (4)
M x (t) ("4 , E(X)
:(.1-%•11
ii=-12=15002= 1 1500
-x e l500 (x) = , x> 0
:P(X>3000) & y_gliziAll (1
..• P(X > x)= Ot
P(X > 3000) = e 1500 = e 2 = 0.135
150L)c- .iaJ. b.j..c. „53.,., 150 ja& (2
L19c24
150
P(X .5_150) = F(150) =1— e 15(X) = 1— e--° =1-0.9 =0.1
: iLui L5- bsui L,1c_ j5,14.1.:i ...A (3
1500
- 123 -
P(X > 300 + 1200 I X > 300) = P(X > 1200) :L.4i
:(3-2-4) Lrl-i-g
stiai zitItA uts. 4:21 j_54.13 Lria.4 4:111 X L.5:11. 31 j.!•;1.11 Citc 13)
:Z.1.11/3141 kit'l_C11
f (x) = 5 X 0,.441S a
0 , o.w
;'•
Li
(2) ,a L:1341f111:t..,95 (1)
(3,11. 10 (53) 5 ahl(c) L331.b 10 0.. j.'S' I slIk 5 u.4
(4),-.44.11.4.11k4111411 (1,3.1 j1...i...411,41)..,3713 c ø (3)
.X 03.17).11_5.411
a = A, = 1 5
C.).4 (:)C..4y.
- 124 -
u---1 215
P(X > 300 +1200 X > 300)
P(X >15001X > 300) = P(X > 1500)
1500 -1 1500 e e
300 =
= —0.2 = e0.8 - = 0.45
e 1500 e
:3 7uS13.11 C37,11j
1200 4
P(X >1200) = e 1500 = e = e-" = 0.45
P(X > 300)
„.9.1 .5411 j..4.eiti ..1
CO
op CO
f (x)dx = 1 =Jae 5 dx=a
.
= —a5 e-1 = —a5[0 —1] = a5 [ o
1 5
F-1
;
P(X < 5) = F(5)
:P(X < 5) 311,110 (2
J(5) =1— e = 1— = 1-0.368 = 0.632.
e 5 1
r7-1
:P(X > 10) L4.911--11(y)
P(X >10) -= P(X =1— (1 — e 5 ) e-2 =0.135.
:P(5 <X < 10) L,-1_311--1 1(E)
P(5 < X <10) = F(10) — F(5) =e 1 — e- 2 = 0.368-0.135 = 0.233.
(3
2 1 2 n E(X) = —1 E(X)= 5 crx = (Tx =.Z.J x = 5
"371 -4,..11 I'd all (4
1 A M x (1 ) = M x (1) = t — 1 — 5t — t / 5
(t) = —1(1 — 50-2 (-5)
AVA, (0) = —1(-5) = 5 = E(X)
r•-•
- 125 -
1( Thu 2
.1271.1 e 2 cr dt :LA 0-451,;11 F (x) = 0 a_114 (3) Li
_1 _ u--=-1215 -;iS::4
(Normal Distribution)
-1....3.5.Z (3-4)
(1)
u.5.113 ;1_,4 I Lci *A41. LJJ
t4,-- 11 .21,-‘1sk+,11 ui
1 4-4n 1+-1.5.5.3 '171 u -Ltn .)-.1k9 SJI)-'31
A--?-J-1-.1 c.'.1:1"*" &Ball L.53) 1-Pui-31.1
:(k.s.1.1.441 at.S11 4-4),L1 (2)
0- 5 P ti.:19131 i c_t_9-c:,. X J -4-11 .r4-9-'- J'
•sjy...11 ,s1c. z.431.4141 Alits
1 Ax-12
f (x) = 2 a
cr c. — CO < X < CO
,r =3.1416 , — oo < X < cx)
:c.)1
0 < < co
3.0)1 ,D14...{11 L53 k7,1i ta0 J p
2 (:)414.1.3 11,3:5144 ( L ti:13:3* Ai X J.17.1.11 C.J1 (sic. :16.1.111 X — Att,0-2 )
o-2x = 36 =
X — N(4,36) = E (X) = 4
171
Li Li
Li Li r - I
:4,14 4-4.53c. 4...9:1 *A-114 (4)
cr22
M (t) e Pt+ 2 X
- 126 -
Li
-1- L.31--.3.1 L t215 J
: u:c22-) (5)
E(X)= ,u : jr6 4.,6.511 (1)
X ap.A JA2. L13., .(5.44.1.11 t.3.33:111 efl
431 yak, t_zio latb L;1.e (1).4 _)LJI
.511 = J131.11 = C.J1
j.4.1:1U
f -\\ •
P2 P3
p X i4 atag Gr 2 t.3.33:111 L: ),41 (2)
62
(x)clx =1 4.3LLI Le y„?.kill L5.3_14,11 cua:11Z.1 ,-11 (3)
(:).4 c.;114:N JIA3 cci '4).41431 f(x)
(4)
L.j 9"?'_,9-4 :;k71, A-421-&-11 .f(x) U3 4, iQ a
.(T11143 ,,4113 LIS.1
(5)
:t4i (6)
- 127 -
-1- di-A: 1 215 siL-4
f (x)dx =1cji c.:14S1 j _3:111i : (1-3-4) LItA
I = f 1 6.- ` ' dx 2rc
dx = — :.I= —e, dx a „42ic
1 I -12 z2 x - p = e dz , z =
liTz- , a
1= 1-2 0. t fe-ii2 dz 0 112g
z2 43.4 4 U e-T c.JC47 )
z2
:N=— L:p11 2
1 dN Z = JTV dz = -(2N) 2 2dN =
2 -‘12N
2 I N -2 e-N dN ..12;z- 112 •'0
F(Y,) = . = =1
-‘17-c 1-2
-Viz
(*)
(*)
F (a) ya e-bY dy = , r(y2)=1.1Tr ha
dz = dx
co
- 128 -
Li
1 JUia.1 215 s.9.11 Cji71.)111,)04:1U A.4.2•A
E(X)= ,u Var(X) -= o - x2
:(2-3-4) LIV:LA
L.c..ii jiil M(t)
•c3—%11 L'
t 20-2 12,72 Ad+ ,ui+
L M X (t) = e 2 , (t) e 2 .(1i + 0-20
t20-2 cry /11+— pt+—
A d (t) — e 2 . (00- 2 ) + t).e 2
E (X) = M (0) = 1 x + 0) = E(X) =
M (0) =lx cY 2 .1) x 1 x = o-2x + 1.1 2
• Var (X) = M (0) — ( M 'x (0)) 2
2 2 2 ax — ,u 2 x
.* . ar (X) = x2
:(3-3-4) Ji
. Y N(ap+b,a2o-2 ) L.-.14:313
Y=aX+b X — N(u, 0-2 ) u15 13)
( 20.,
1,1 M x (t) = e 2
M y (t) = ebt M x (at) u2t2a2 aycr2
fiat I b(-1
e .e 2 e 2
a212(72 (up+ b), +---- --
=e 2
a 2a 2 ..9A 4443 (t JAI—) op + h & 4 In 9114 jx..11;4.9.4
t 2
JA")
:LALA :41.S!)LA
- 129 -
1,3.6.421-1
4-2,4121.-13j:I X Y = aX +b :Oa oi 4.1jt..411 c).4 6.7=1
c1 2 0-2 (...,Y.In4.9 a + b 1:14-4 :91411 44-171-1-12
. Y N(a,a + b, a 2 o- 2 )
: 0441.:§1. 0.2÷.11:111 5-11.9131 (44)
(Standard Normal Distribution): :ZAss*L4.11 (1)
JjL 4.33 = j 11=0 44i 11.1...!,!1,11c_-11.2_133:111ata91.1.1 tii9:21
4 4.1) 4 cz_11..,133:111 43.4 11-;11.1.4=.1
(4j1,1?-42,11) LeAtAl J.1;1,11 LA X J.I;.;
El
0
E(Z) = 0 , (3-; =1 : L-3.
atiii A21.1 41,5 (2)
cji_c3z Aito :01.1 cji x
2
- JC(Z) =-: r e 2
-‘1271. — oo < Z < co
1 F(z) = P(Z z) = e 2 dt
,11271-
- 130 -
P( Z -z)
Jj1.1. 3 z L5.1) — co b., Lc.,44.31 Ls a,14.11L 2O j1.0,1 L ju-i41 13A3
4 1:1tt
,LAjisd JAi;th.1.11 (4)
(.32) L5-171-110.1.. d-).5-1:1 (5)
x — r 4.4=31 t_i3A 43 Z - I' u13 X - N(1.1,a2 ) J.;;",-11 c j1_C 13)
a A,
Z23t.:31 Lci (3314...11 z — N(0,0 (,<•13
x x (i) P(X 5_ x) = P(X - P
- ,u x - Aix )= P(Z )
x
(ii) P(X x)=1- P(X x) =1- P(Z z)
(iii) P(a 5_ X b)= P(X 1))- P(X ..•C_ a) = P(Z z1 )- P(Z z 2 )
z =b
x
a- Aix
(6)
:(1-4 -4) LILIA
2
•C' M 7 ( t) = e 2
c,e,41 c5_,y41:11 j5-311 c2J.0
Li
- 131 -
r - :
x M
-1— j- u215j.cC C.1171.5.9.1. jAcIU 4
Co :2
Z/ f e .e -Tdz Mz(t)= E(e )= ,
co e
-yz dz
,
co 1 1 —(z-t)2 =
-12
e 2 e 2 dz -Co
y=z-tdy=dz
1 °3 =e 2 r---- e 2 d
-427r , i2
:(2-4-4) Jewa
0'212
t) = efit+
2 N:".1 :61.111
:(.3=11
X=o-Z+11
M aY +b (t) = etb M y (at)
• M x (t) = etP M z (at) 20-2
tp+t 2cr 2
= etP e 2 =e 2
:(3-4-4) dtia
CJA Z N(0,1) (:)1 13) (1)
i) P(Z <1.25) ii) P(Z -1.25) iii) P(Z >1.67) iv) P(-2 <Z < -0.2)
- 132 -
I!'
CLJ
L.,
-1- (.3u1=-1
i) P(-8 < X <1) ii) P(0 < X <10) : X - N(2,25) L'jlS 131(2)
P(p - 20- < X < ,u + 2a) • X - N(ji, )
P(Z <1.25) = F(1.25) (1
(1.94.11 j&.1:1 4.11A tr5-tiI JI1 ajØ Jj JAlatb .4.47
0.8944 sc-bill 0.05 1.2 1.2 J1--Al
(i) P(Z <1.25) = 0.8944 (ii) P(Z -1.25) = 0.1056 (iii) P(Z > 1.67) =1- P(Z
=1-0.9525 = 0.0475 (iv) P(-2 < Z <-0.2) = P(Z -0.2) - P(Z -2)
= 0.4207 - 0.0228 = 0.3979
: px =2 o-x = = 5 1-1.1.-11 X _.),,,Ax.3 (2
P(-8 < X <1) = P(X <1)- P(X < -8) - -
P(Z <1-2
) P(Z <82
5 5 = P(Z < -0.2) - P(Z < -2) = 0.4207 - 0.0228 = 0.3979
(ii) P(0 < X <10) = P(X <10)- P(X < 0)
- = P(Z <
10 - 2 ) P(Z <0 2
) 5 5
= P(Z < 1.6)- P(Z <-0.4) = 0.9452 - 0.3446 = 0.6006
= p _9 o , = a X j i (3
P(X < p + 2o- ) — P(X < — 2u)
P(Z < p+ 2o- —
) P(Z < o-
= P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
•:• • - 133 -
C1j1.,1
(i)
C.1171-51.1W•i- 1-1
:(4-4-4)
M y (t) = e3t+"2 0-.11_,A31 41.11
py, ay , P(Y > 3), P(-1< Y <5)
0.2,2
C.5.j-1 ..)4:114 -1.413n Lsk5 Mx (t) = e l" 2
py =3, 8 =—o-2
o-12, =16 =4 2
P(Y > 3) =1-P(Y <3) = 1-P(Z <3-4 3 )
=1- P(Z <0)=1-0.5=0.5
P(-1< Y <5). P(Y < 5)- P(Y <-1)
= P(Z < 5-4
3) P(Z <-1
4-3
)
= P(Z < 0.5)- P(Z < -1)=0.6915-0.1587=0.5328
:(5-4-4) JelA
15 4.3L9. (cJ A 4).31, 100 01S3I1 tLialS 131
3 70 ap. (3) , 80 D.. (2) 4 125 (:).4„..1c1 (1) :..).) .433 0.13.11•14,_,L11...i
. 130
:33-11
:rt..4.1t11 L4_31.,11 X j.!;1.11. 4Sall '4j.)1
,ux =100, ux =15, X - N(100,152 )
:P(X > 125) L7,_+_711--11 (1
-1-
Li
- 134 -
-1- JJA:21.1 4
P(X > 125) = 1— P(X < 125)
= 1— P(Z < 15
125 —100 ) = 1— P(Z < 1.67) = 1— 0.9525 = 0.0475
0.0475 (100) = 4.75% :L5A J.I/L-411 A !.
:P(X < 80) L.,131.1 (2
80 —1 °°) = P(Z < —1.33) = 0.0918 P(X < 80) = P(Z < 15
9.18%
:P(70 < X < 130) L:4).11(3
P(70 < X <130) = P(X <130)— P(X < 70)
130-1007010° P(Z < P(Z <
15 15 = P(Z < 2) — P(Z < -2) = 0.9772 — 0.0228 = 0.9544
95.44%
Gamma Distribution :1-04 471.,).9.-.1 (5-4)
- 1
c:-131s 13 ) (n, /1.) t.is.71. X jo1
:y.11_4:3 I
(x) (n)x' e ; x>0 , 2>0 ,n>0
10 ; o.141
- 135 -
Li
:
-1- J siSla ..L.)44. -9:1 .'il71.5121,).`=Ci
JAI.S5) J :k.1b L.J.4 3_4g
F(n) = x"-1 x dx
udv = uv - vdu j ,ccg
El
U Xn-1 & dv = e-x dx
lc j du = (n -1).xn-2 dx & v = (Di
F(n) = r xe _xdx = {xn_I _ Jo ( —e -x )(n -1)xn-2 dx
= 0 + (n -1) xn-2 e-x dx
= (n -1)rin -1) = (n -1)(n - 2)• • -3.2.F(1)
CO
F(1) =Ix° dx = [-e-x 17 =1
F(n) = (n 1) !(.jli
(i) 1-414 3.41.511 E.1111
Li
1.
2.
3.
r(n + -1 ) = 2
r( n +1 ) =
1.3.5. • • •.(2n -1) "Tr , n = 1,2,3,• • •
n = 2,4,6,8, • -- 1.3.5. 2"
• - - .(n -1) jr . , 2
1 r- r(-) = ••oz- 2
n 2 2
- 136 -
_1- jt_Aa_,- 215 J A ...).4.41:1
ya-1 e-by dy _ F (a)
ba 0
A" x F(x)= ft"-1 CA' dt ,x 0
F(n) 0 cok 4-27.13,91 — 2
M(t) = ( A — t
;2>t 4.A13.1.1 ;41..5.4.11 ;L11.111 — 3
,Lt= E(X)=-1-7 .
11 0-2 = = —22
. it: = E(X1) =F (n + r)
r (n)
. n=1
: 449:1 — 4
:jk 441.116 — 5
:ZAlk t:.•1 .1+.)La — 6
: 4-4.33c. 43-31 JlJ 31 0)
cr,211 tag (y)
:C.)1 1-44 &..1911 -:(1-5-4) 3111 4
. UJL L J fix) (1 )
(2) _ F(n + r)
r F(n) (3) /1 = " (4) a
2'7
— if , (5) M (1) =(— 1
) fri , (6) r(-i) =
• (.3-%11
g5A (n,2) (:). ..„1„11_,, j.!;1.1 *1';c11 :1,31 (1)
f (x)= xulH e ; x>0 , )L>0 ,n>0 F(n)
; o.w
- 137 -
-1- Li\-• 215 ".._)-C3-.
cLiL. ZU it (31.4:]f(x) :LIU
(1) f (x) 0 Vx , (2) I f (x)dx =1
:_,t0 Ls-112 -12_A Z11.111 cj.4 c., _9
n co
f(x)dx = xn-1 dx =
f xn-ie-A.dx 0 r(n) r(n) -10 (1-5)
177 Li
dy = Adx L1415.111 (1-5) Za..All cci =
:
An j_eYn-1 -y 1 1 I r(n) F(n) 0 F(n) 0 2n-1 A
dy = r(n) 0
y e dy = r(n)
=1
. Y91"11 Y
= E(Xr)= f xr f (x)dx =
13 An n-1 -Ax = f Xr r(n)
x e dx = An x) f Xn+r-le-h dx (2-5) F(n) 0 o
a=n+r, b = r •
Pr dx = 2" F(n + r) F(n + r)
F(n)i0 F(n) r 2T (n)
= r (n +1) nr(n) n 21r(n) (n)
LDI :c2-)631 C541.1.1 (3)
- 133 -
2
v2 --•V271- e cly =
co y2
1 e 2 / e dy = NETT'
CTC",
-1- L11--1 u^-1 215
J1 jyU .14.=.41
14= E(X 2 )= F(n + 2) (n + 1)nF (n) n
2 n 221-(n) 22F (n)
a2 = E(X 2 )-(E(X)) 2
0_ 2 = n 2 + n n 2 _ n A2 -
:Di -1,<3 1-4 ( (5)
An - • 11'17 n-I -x A-t M x (i) E(ex t ) = ex t x"- e = x e )clx
o r(n) r(n) 0
F(n)
F(n) (2—t) (2 —
Yill="31 JJ
cj_c_ . L41.61..) (:),A (6)
F(1) = -1 dx x e ' dx (3-5)
2 0
ydy = dx. J.rv6 ciALL11 Di -1?-3 (3-5) 2
Y2 2
_
Y - F(-1)= —e 2 ydy = i J e 2 dy
2 0 2 2 0
:ui
CO
_Y j•Tr r- F(-
1 )=Jje 2 dy = v2 = 2 2 0
- 139 -
-:(2-5-4)Jtla
(:).4 J.Ac• 3-4-j X j.;;1,11 LDLC
'L:43 JISia C f (x) = exe-2: , x> 0 :4-4b
.1 (2) • r C c2-3-41 *A-4 (1)
L:).4) J1-41z.1 (3)
LI) I J.LIL? • :c.d.:a...a L;a1...Lzi j t.L j_i.1 X ui c,..:01..3 (I )
A = 2 , n —1 =1 n = 2
An 22 c = = 4 ZIJUi-4•11-.1 a-?:1 47.11 -J
F (n) F(2)
[31 :
f (x)dx =1 cxe-2x °ix =c F(2)
= c I = 1 = c = 4 . 22 4
J11-,11u j j :7‘.7X, (3)
0.25
P(X 0.25) = 4xe-2x dx 0
J udy = uv vdu gpAi c.4.341-1 L;,a-111 13A L711-,AL
- 140 -
ril
-1-215j- OL,1)1.1. lihdoU44 ..!
u = x & dv = e 2 dx
du = dx & e-2x
4.11P J.!;7; (-1.°1-L11 DIJ v = 2
0.25 0.25 --2u 0 2.25 _ „ , e P(X 5_ 0.25) = 1 4xe-2'
2 dx = 4 Judy =4{[-u - ]0025 f
2 du}
o o 0
0.25 0.25
= 2[-ue-210'25 + 2 f e-2"du = 4-0.25e-115 J+ 2 f e-2" du 0 o
= -0.5e-" - [e-2n 100.25 = -0.5e4.5 - [e4.5 -1] = 1 -1.5e-°.5 =0.09
Chi-Square Distribution: :sL tJj*-1 (6 4)
Z.1.,..11 (x2) 4LC t4.4 x j.!;•;,11 SA:3.
CS
-:- x v 1 x 2 e 2 , x>
2 2 F( v-- 2
0 ; 0.14)
v
: 4.21,91*— 2
: 4:171.14:1 — 3
= E(X)= v
0-2 = V(X) = 2v
.!-
"(1) = (1 2 ; 0.5>1 4..4,93.•LI811,9.431:;t11.1.11— 4
- 141 -
El
ri L
-1- LiL.1..1 L.-1215 -o_)L-.
c:11.2Lz.L.4_ 5
. n=—v , = 1 — ZI.c• C.)-a J4.71. t„,_335 (1) 2 2 L -3
cD1 ,5333 L 11% csi X (.5:31,,L2-1. I 3;t,11 (i‹.1 "bac (y)
. X — LA1131 i'4.):13 V ..)"1.3 4.1s
-:(1-6-4) delA
f (x) = x e 2 x> 0 *A41.4:111 ;;Li'LL.C.11 &.y. X Lsili2A.11 _x:•121 ul.S 13) 16F(3)
. x J.11-11 ra3=- al.L15_9 &9_9:3 CJALJ jj Y311"Ili
3-%21
i f (x) = 1 x2 e , x> 0 J-C-2-1 f ( X ) LDS" -‘2113 _9 X r(3)
: L..11 4:1 ;1-1.)-1431-.1 4211-i 4 41c _9
V =3 = v =6 2
E (X) = v = 6 & cr 2 = V(X)= 2v =12 ,
f- 1 M(t)= (1
2t 1 —2t ) 2 = ( =(1-2t 3
1—
? jt..3.11
- 142 -
-1- (.3.A:L.,• AA—NA
Beta Distribution: :V-;71-1.. t.T3..19:1 (7 _4)
(a ,b) t-.9-.1 X _1" —11 L5i1.3‘:"11
Lrit Aas-tfis
{ f (x) = fl(al ,b) o
x"-I (1 k)b—I ; a>0,b>0
0.W
F(a+b)I(1—x)b-1 ; , a>0 ,b>0
• Li (x) = F(a)F(b) 0 ; o .w
F(a)F(b) fi (a , b) =
F (a + b)
J-) J 1;1,3 .Lj_5:111
13 (a , b) = xt" (1 — x)" dx a>0 , h>0. 0
= E(X)p(a + r,b)
fi (a , b)
it = E(X) =
0.2 = V (X) = (a + b+ 1)(a + h) 2
(lb A.4132.! D 42,54 .1k:u1,)-C:111 .;1;.:111 2
jib a a + b
ab
- 143 -
215 7-, J.L-4
;441.:4 —5
x = 0.5 J_PJL44 f(x) uj a b 11-11-'• L5i
(0 , 1) t:3.19:11 f(x) cji-C:i a = b=1 (y)
a -1 . x, = j f (x)= 0 (si--s (3:131 -.::;11 3.& : xo I 13A (-3133-4 (E)
: L5A .;‘7.1" 1521
l o x <0
1 F (x) = (1- t)b-1 dt , <x <1
fl(a,b) 0
1 , x
Incomplete beta function (Lai' J71.=.) F(x) L
. X Z47,11.11 171-4.-1. 4:111 X Ca+.1:11÷.1 .L"=11-&• d_91•1 1+1_9
C4i &.:);92. -:(1-7-4) c3114
zji...5 fix) (1)
(2) pi fi (a + r,b) = F (a + r)F (a + b) (3) a , (4) 0_ 2 = ab fi (a, b) F (a)F (a + b + r)' a + b (a+b+1)(a+b)2.
4-43 &;:!. X J.1;1-1 ;141.1-.41,* VI (1)
f (x)= 1
xa-I (1—x)b-I ; 0<x<1 , a>0,b>0 fi (a, b)
0 ; o.w
41142=1 rtit."15 Z113 ji.4* fix) ZI13 ul IL,21
Li
- 144 -
f=7)
215 -*a_ja
(1) f (x) 0 V x , ( 2) I f (x)dx =1
_96:s L_Ali Lai .L lL J J
1 1 1 f (x)dx = x"-10—x)b-idx= f x"-' (1 x) 6-1 dx =
0 fi(a,b) fi(a,b) 0 fi(a,b)
. y311-11 JAJ
= E(X r )= (x)dx
= Sx' xa-I (1 x)b-1 dx = 1 xa+r -1 (1— x)b-1 dx
o fi(a,b)
fi(a + r,b) = F(a + r)F(b) F(a + b) = F (a + r)F(a +b) fi(a,b) F(a + r + b) F(a)F(b) F(a)F(a + b + r)
JA3
F (a +1)F (a + b)aKa + b —1)! a =
: oNc.1 Z9--)63 **.L.9441.i (3) F(a)F(a + b + 1) (a —1)!(a + b)! a + b
. Y._91)=411 _03
: -1?3 '111:1 (4)
a 2 + a 1121 = E(X 2 )=
(a+b+1)(a+b)
o-2 = E(X 2 )—(E(X)) 2
2 a 2 + a C72 (a 2 + a)(C/ + /2) - C1 2 (a h +1)
o- =
(a + b +1)(a + b) 2 •
Y_51-13.431 JJ
- 145 -
0
0
(a + b +1)(a + h) (a + b) 2 (a + b +1)(a + h) 2
ab
-1- u---1215 ,3,1 jo.c21.:
X ,..;11. -;(2-7-4) jt.4
f (x)= c(x — x2 )" , 0 < x <1
: CJA (.3S
(1) c , (2) ,u , (3) o-2 , (4) xc, , (5) P(X 0.5)
: -LI f(x) L. LDI
f (x) = cx°3 (1— x)" , 0 < x <1
]-(a + b) (1) c = , a = b = 1.5
F(a)F(b)
F(3) 2! 2 8
3 3 1 1 1 r— r---= 7t
,c=--r8 = — c
r(-2P-2) —2
r(-2)- 1 —2r
=
1 VI lr'‘Iir
a —1 1.5-1 (2) x0 = , = x
° == 0.5 .
a+b-2 3-2
1.5 _ (3) ,u = E(X) = a
b ,
a +
ab 2 (1.5)(1.5) (4) -.• cy2 = =0.0625 .
(a+b+1)(a+b)2 3 (4)(3)2
(5) P(X 0.5) = 0.5
. x=0.5 t?...15:311 (..114 (4 LU J
2
- 146 -
' Js jo4.1
(8-4)
4.5dt, 4:ATIS Z113 V31.yfk. ..)4ia..* X DIS 13) .1
x O, >
M x (t) = (2, :c.,43 X j 11 .;%. Lji
Var(X) , E(X) ÜAJs
A M y (t) = ZIL 43 y L5.3ii,t=11 j.!;.;.11 ut 1:))
. p(2 <Y <3) f(y)1. Z11.)
cssi Y) Z = 2Y Z jii 2J1.111 c .,1C1 .3
Z j2;1 11 F(Z) j f(Z) DA LS
f(x);;411-4:1-. ) 1.6U-C 11,21-) 33 X cpjiLc'' ‘2_93a 1JA."A-11-111 L-22Jc- .4 <X < co
: 1311.-411 M(i) = e31-1-8(2 Y cey.-.L1-11 ..):1;1;-11 ..)x-13 S-11_9-4.11 ::\-11 -111 .5
Y .J4i1.11 jt ..•^ 11 LJu?1-.1.3.1 tki-111 (DA JC -.)1-?7,..) (1)
— Y-3 :Di t4.1 M(t) f(z) is (3) P(Y‹ 3) 4
(2)
X ccslyte- z1L) f(x) ;dali .6
(x) = b — a
, cr.X.15_b
M (t) = —ea" 4.A X j4;L:1.411 r _53.111,3.1_,..1111_11.111 Laifil
t(b — a)
- 147 -
ri
-1- J- J i2l5 J 4 JA.4L1
1 2; j.S1.1.11 JIZAi j4.c. (1) .7
t4.3- ti Lsaii C j i1 (:).. 13) (2) 150 -1-.-..9=1-4-.)
300 LD.-
LA415 ca.:J..141b.A L1Ji
a2 2 g+—t E(Y)= p cu cz-3-.131 M y (t) = e 2 : Yj-11 e 3321 s43-11 4.al .8
:CS-4 (3 24 cr4 L* M x (t) = e21+4512 e3.3,d1 all 411 .9
ax , P(X> 2)
. :4411-4a.H •AAlfis Lr:141 f(x) (1) : Lt .10 .10
(2) p =F(n + r) I-- 0 p=—n (4) 2
=4, (t)= (5) M (-2 )" , (6) F(-1 )= 2T(n) 2 —t 2
rk.11.1 L j L.Jiy.A j.i4 c.,y1_,....411 jiA,3X j.2;1.11 cil5 13) .11
j1.14 c f(x) = cxe-2 x , x> o
4alt.'15
r
? 0.2 .f c L:13-.112 :t4i (1) Li
f
:cu JS X j.241-11 c+1-,,,- 13 X 2,2 cg 1.-1 .12
Lta.21,..3.31(L,J) Lc tfl e3.31(1)
. :Ln (.Siz f(x) (1) : cuuy.4&12.3*-11 .13
- 148 -
-1- Jul-1 L 215 bJ4 ejt,u111.
f (x)= c(x — x2)" , 0 < x <1
(2) (a + r,b) = F (a + r)F (a + b) „, a
p (a,b) F (a)F (a + b + r)' ) = a + b, (4) 0-2
a(a + b +1b)(a + b)2
; 1:kiUS Z11. X 1_32.3. I _x;i.11,DLC .14
(1) c , (2) /I , (3)a2 , (4) x„ , (5) P(X 0.5) .
*A.3l2.1S ..L.31.1.3 41.S t4.).4 t.1.3.35 t.1..1,i11.j.ifk. 1 .j4;t14. X ulS 13) .15
(x) =1
— x7 e 2 , X O.
8c
.)21 2J1-111 (3). V(X) D4--.1:1113 E(X) (2) . c Uifl Z4.475' (1)
• MX (I )
:A4L.S.I1 Ail ';<11 4_1141 olS 13) .16
f(X) = c X2 (1 — X), 0 < X 1.
X 44.11 c (I)
El
Li
215
c.J..ca.11
Joint Distributions :k-Sji:‘,411 (J12.4.),.9:th
-;:t..4ssi4 1 _ 5
?1_11 csi L:17-1_14JS. ) tic. J.1;1,1 Ji
Cy, L.5-=-11-6 , Univariate Distribution 4-1..t21
(4..th Z,4),111u1 j I
c_1131.5 1:41 I ;A....a L3 ,
JAc- 0_15 Lb]. )
k?•14. t34 LIJA. ,A11:14.) (X,Y,Z,L) Lerli-45-3 i:).5-y..1 -1.3 cL (SIC' Zii2-4
Lc'fl C.)34 A7•14. we J d-Cij U(X,Y,Z,L):C2 -->914
s w u(w). lx(w),Y(w),z(w),L(w)} :34 11:ci.::;-11
.1.3j 131-i, ku..1-& 3L tz.1_3.35 X,Y,Z,L c-.31 J.:;1-11 ._)4 Li-C1
:1-IA 'Li-17.310 L5-A,A:373-?.1 L-116 .3.35 (sic. -' 1-131-iLA jti j;
jj Joint Distributed Random Variables ,A1.1-t-11 't113
t:4_1,111
Bivariate Distribution (4 11j1.11 ,A1,j;t-11 t..)3111-1(:),Lx;1-1,--41f-t-11
Trivariate Distribution (.51=1 tthall-.3IJA Lk 45
Multivariate Distribution j't"11 t:•3 _3_9111 LeL":1,3
• ()A:LAA i&N1.3 tlalq.L4.- 16 _31 zu.,31,41Sji JAI A (LIA).!;1_,
.t3:111 _}..4.1 0...u.:;1.11_3 LA121 j,t.11 (sic. 11%
Bivariate Distribution0-11-121 19:111 2 — 5
. U(X,Y):0 ---> 912 L5i59. 13A U4X,Y) c5:11.A23. 4:'.."1/ 411 CJ1 0:43Ji:11
- 150 -
,••••••,
C.7
cyz LI-C (X ,Y) 11 t..9:3 (-.-Lul'3 ...'1^11 141
Joint Probability Function : k 1-2 — 5
kg \ 4<II (x,y) DI j.!;1,11 ulS 111
[1
:c5A ( (x, n
cp-13 6'4:U1 (..)..9
gc&
f(x,y) Z11-111
f (x, y) = P(X = x,Y y)
X •-= X2 '" • n Y = YilY2 ,• •,
-1- 3 215jLi...L..1
f(x,y) .Y, Y2 • • • Y; ... ... .Y„, I f(x,y) Y
X1 f (xi , Y1) f(xl,.Y2) ... ... .f (xi , Y J ) ... ... f(x,,Y.) f(x 1 )
x, 4
1
f (X2 3 YI ) f( X2 5 Y2 ) • • • • • • .f (x2, yi ) • • • ... f (x2 5 Yin)
.
f1(x2)
. . . . . . . . . . .
•
ix,
1 a
.f(x,, Y1) ./ .(x,, Y2 ) ... ... f(x,y )
.
... ... f(x,,Y„,)
.
fx(x,)
.
14 :
. . . . . . . . .
.
• .
.
i
. . . . . . . . .
.
, x„ ,
.f(x„, y,) .1(x„, y2) ... ... .f(x„, y.,) ... ... f (x„, y,,,) 1:,(x„)
i ' Z f (x, Y)
,
1
1.),(yi) .1y. (y2) ... ... fy(y 1 ) ... ... fy (y,,,) 1
- 151 -
L.
J..11
-1- u---1215 dY 4
crLuL" f (x,y)
(i) f (x, y) 0 ,V x , ,
(ii)1 f (x, y) =
1.611C-.1 3 7" c-D-4:1. 1 r _5:kJ (-31'1"'i a.1643.:" (X ,Y) i "7'
9q2 Ls_31--11 Li
f( x ,y ) 1131.1 (LJ• a?..:3 1313
(i) f (x, y) 0 ,V x , V y ,
(ii) f (x, y)dyclx =1
,y) 2J1. Lr f (x, y) ui
Joint Distribution Function : :4-s111.411 jj A-114 2-2 — 5
01-,;=%3 L5-1c- (X,Y) (jtS 13)
F(x,y)= P(X x,Y y)
(X ,Y) 4J1' t-11 ULLr42-1
: (-71- 41- F(x,y) ZA-.0
F(x, y) =
x Y E f (u,v)
f f f (u,v)dvdu (1-5)
LJ
- 152 -
-1- -1-,1 215 ji U.s.‘11.1
: F(x,v) L)=1.3.1. .U;=4.1-) L.
(1) F(x,-00) = 0 , F (-00, y) = 0 & F(-00,-09) = 0
(2) F (x, co) = F (x) , F (co, y) = F (y) & F (co , 00) = 1
(3) xl < x2 y1 <Y 2 = F(xyi ) F (x 2 , y2) •
. F(x,y) jS 1:1A
Li (4) P(x, <X 5_ x2 ,y1 <Y 5_ y2 )
F(x 2 , y2 ) — F (x 2 , y l ) — F(x, , y2) + F(xyi )
(5) F (x+ , y) = limF (x + , y) = F (x, y) & e-40
F(x,y) = limF (x, y + s) = F (x, y) . 6-40
. u.2 ,.211 F(x,y) 3 c. 1:143 .43
Marginal Distributions
421 .44**1411L.t JJ 3-2 - 5
it u-a-& t:3.151 (x,Y) CY3J—\r
(:11I)
çc j",t,11 (441÷.1L ;*A31_,_11
t_53L f(xcy) aa11ü f(x) L,_JL,•,.
r'47.4 .f x (x) = (2 - 5)
(x, y)dy
: LD1 LY-L.1.3
- 153 -
-1- 2-.1..1 u--1215
fy (y) =
F(x,y) 1j4 f(y)
fy(Y)=—dy F(x,Y) -3 f x (x) = —d
F (x, co) dx
d"=1:." (X, Y)
2
f (x, y) = a F(x, y) axav
ti3 F(x,y) JL F(x) Z1LD
X CO
E E f(t,Y)
F x (x) = F (x, co t=-coy=-co x co
f (t, y)dydt t=-00 y=-on
(3 — 5)
: jtait-.3
.0 Y z f(x,t) x=-001=-co co y
f(x,t)dtdx x.-00 1=-03
Fy (y) = F (co, y) =
:ciA.)41
(X ,Y) u 4j1 .
- 154 -
-1—
Z111 4:y X 1431 t....33-..111 :;L113LJL 1 c •Ail._]1
(1-5) .A:s")631 1
F (x,00) = P (X x,Y < co
f(t,Y) I.-00-00
= , f(1) = Fx (x) .
,Y) 4.**LAII ut--
1.11,11.,,* :k.S.5;:-.11 Z11311 (:).. X J 41 4j ..,033.11 401.) 0.3A 4)
l_AS (1-5) 1.11
x 00
F (x, 00) = P(X x,Y )= f (t, y)dydt
= f f (t)dt = FA (x) .
Joint Mathematical Expectation :(21j-LL-41. 0-(:=2171)11 -1..1-1.11 3— 5
A411,:.1 11 kth 4311 (x,y) u_i 1613 4,5A g(x ,y)
g 4J i JiQ43L.1 Lkyyft, g Z11.1.11 t...9_9111 /(x, y)
t t.9".111 tz1.. j (X , y)
aur;1,11
IZZ g(x, y) f (x, y) , y
E(g(X ,Y)) = <„ .
J$ g(x, y)f (x, y)clydx
(4-5)
L7:19 _p jic u.4 4:ji JA3
- 155 -
-1-J'-..1 215 _iLIA
: ci(x,y) 4.;22-:1
g(X ,Y) = X c:-13LC131 - 1
{Ey, xf (x, y) = ExE f(x,y) = I, xf (x)= E (X) x Y x Y x
E (g (X ,Y)) = „a if xf (x, y)dydx =1 xi f (x , y)dydx =1 xf (x)dx = E (X)
Ls . e1-4411 &3•111-.' t-s_5.331 .44 DA J 5 L:JA LD L i
. g(X ,Y)= Y . (Marginal Expectation) X _),:41-11
:J i t_53 g(X ,Y) = (X - x )2 cL1:11S 13) - 2
EE(x f (x 3)) =Dx Aix)2 If (x,Y) x y
=1(x - ,ux )2 f (x) = E(X - ,u x ) 2 =
E(g(X ,Y))= 11
(x - ,ux )2 f (x, y)dydx =1(x - ,u x )2 f (x, y)dydx
1+11 U i_35.11 L. cJ J4ti CJA c.135:3. L.D1 -I-
. g(X ,Y) = (Y muy ) 2 11) 31.31.)j . (Marginal Variance) X j.141-11
: (1-5) S-Lia
i3-14 csi 1-4-C Lc& ( X
Z11.111 c_.- L'ILS 131
; L5.11:111
ti
Lj
- 156 -
f(x,Y)
y
f(x) _2 0 5
x
1 0.15 0.25 0.2 0.6
3 0.2 c 0.15 0.35+c
f(Y)
0.35 0.25+
C
0.35 1
in
IT :1 413_9
L5113 C (:141 Z4YJ .1
C = 1- 0.95 = 0.05
215 SJC-La JC.J17?-5:0-)441-1
f(x,Y)
Y
_2 0 5
x
1 0.15 0.25 0.2
3 0.2 c 0.15
: Zoti8. : ‘7-'_91)="31
. c c2-41 (1)
• fx (x) .fy())) (.31 .3-111 (2)
V(X), V(Y)(4) E(2X —3Y) (3)
P(X +Y ._1)(7) ,F(1,0) (6) , E(XY) (5)
:d
c.31:111 di-14
Lci LASJL41fl b j J.14 CjA -1j11-c .2
x 1 3
f(x) 0.6 0.4
Y -2 0 5
f(y) 0.35 0.3 0.35
: L.-)_911--431 (3)
Li
- 157 -
215 -1-
E (X) = >xf(x) =1. (0.6) +3 (0.4) =1.8
E(Y) = yf (y) = —2 • (0.35) + 0 • (0.3) + 5 • (0.35) =1.05
E(2 X —3Y) = 2E (X) — 3E(Y) = 2(1.8) — 3(1.05) = 0.45
E(x2),Ex2f (x) = 12 • (0.6) + 32 • (0.4) = 4.2
o-x2 = V(X) = E(X 2 )— (E(X))2
V(X) = 4.2 — (1.8)2 = 0.96 .
o- V(Y) = 9.0475 2
: _91b y311-4 (5)
E(XY) = EExyf(x,y) x y
= (1)(-2)f(1,-2) + (0) f (1,0) + (1)(5)f(1,5)
+ (3)(-2) f (3,-2) + (0) f (3,0) + (3)(5) f (3,5)
= (-2)(0.15) + (5)(0.2) — (6)(0.2) + (15)(0.15) = 1.75
: ith Y_911--11 (6)
F(1,0) = P(X 1, Y 5_ 0)
= f (1,-2) + f (1,0) = 0.15 + 0.25 = 0.4
: 344 Lru"--11 (7)
P(X + Y 5_1) = f (1,-2) + f (1,0) + f (3,-2)
= 0.15 + 0.25 + 0.2 = 0.6
- 158 -
-1- L..11-zi,i 215
j..31 CJI..459.11 JACIU .J
: (2-5) C.ILL4
x y auy.1,11 _.c,t-11 `11 Z11. ca'31.5 13)
f(x,y)=c(x+y) , 0<x, y<2 .
. c i24 (1) :uJi-
.F(1,1) (4) . V(X) , V(Y) (3)
(x, y)dydx -=- 1
:(1) : JJ
2 2 2 2
Li 1 = fie(X y)dydx =-- [xy + y ]02 dx 2 o o
2
—1
c$ [2x + 2]dx = c[x2 + = 8c 8c =1 c 8
0<x,y<2 . 8
-.0.4. L.S ;4,1.411 J13.111 (2)
-.• f x (x)= if (x, y)dy
+ y 1 2 2 1 .fx (x) = f
x dy = [xy +
y = [2x + 2] = x +1
8 2 8 4
x (x) = x+1
, 0<x<2 . 4
: s4
y +1 fy (Y) =
, 0<y<2 4
0
- 159 -
(4)
o-x2 = V(X) E(X 2 ) - (E(X))2 22 .X3 x 2 7_
E(X) - 4x x 4- 1 ch _
(x2 x)dx=743[— 0 4 4 0
11 & E(X 2 ) = —53 V(X) = T.6 .
2 — V(Y)= —11
" 36 SIA11-.1
x y
F (x, y) = f (u,v)dvdu
F(1,1) = fix4_ dxdy=-1 1[xy+2--]o clx
00 8 8 0 2
1 1,dx _ 1,x2 x j
F(1,1) = - - L
8 2 8 2 2 1 — 0) = 1
—8
Covariance :Jall 4-5
-
1-4-1 45-t-11 alL.1111 al X, Y L1,4 J4Wil
1 -g -Jul' 1-446-4.3) au.:;1-11
C".44 j J7).111'111 Cic" 271-3 c x ii Cov(X,Y)
:_3-4 J7)12'111 g(x, y) = (X - E(X))(Y - E(fl) g(x,y)
o- x = Cov(X ,Y) = E {(X - E(X))(Y - E(17. DI (5-5)
- 160 -
'Jojii c:ji AZIA3
o- = Cov(X ,Y)= E(XY)— E(X)E(Y) (6 — 5) x,Y
: JA:1-111 L 2:1 — (T1
(1) Cov(X, Y) = Cov(Y , X) , (2) Cov(X, X) = V (X) , (3) Cov(X, a) = 0,
(4) Cov(aX ,bY) = abCov(X ,Y) ,
(5) Cov(X, + X2 , Y) = Cov(X ,Y) + Cov(X 2 ,Y).
: C.JIA.J41
j5L Jf 4j3
LHS = Cov(X + X2 , Y) = E[(X, + X2 PT] — E(X, + X 2 )E(Y)
E(X, Y) + E(X 2Y) — E(X0E(Y) — E(X 2 )E(Y)
= E(X,Y)— E(X1 )E(Y) + E(X 2Y)— E(X 2 )E(Y)
Cov(X,,Y)+ Cov(X 2 ,Y) = RHS
.Y.91'11 .3A3
(15C11 44
coy _-_-_zEcov(x„ y1) ,=,
COV(X1 + X2 + X3 , + Y2 ) = Cov(X, , Y1 ) + Cov(X, , Y2)
+ Cov(X2 , Y, ) + Cov(X 2 , Y2 ) + Cov(X 3 , ) + Cov(X 3 , Y2) .
(6) V(X + Y) = V(X)+ V(Y)+ 2Cov(X ,Y) .
b.ath
- 161 -
-1- c_11-1,.1
J (:)1145921 jo..azu ..!
-.• V(X + Y) = E(X Y) 2 —(E(X + Y)) 2
LHS =V(X +Y)= E(X 2 +Y 2 +2XY)—(E(X)+ E(Y)) 2
= E(X 2 )+ E(Y 2)+2E(XY)—(E(X)) 2 —(E(Y)) 2 — 2E(X)E(Y)
= E(X 2 )—(E(X)) 2 + E(Y 2 )—(E(Y)) 2 +2{E(XY)— E(X)E(Y)} =V(X)+V(Y)+2Cov(X,Y)= RHS .
(7)V(X —Y)=V(X)+V(Y)-2Cov(X,Y).
. &..14 ba&
•L..111..1=11 J1.4.4 (DIAJ411 :cjIA
Correlation Coefficient:14V:11c342-4 5— 5
V(X), J4 1-m3A c.35 X, Y cil-s13) : — i
Px ,y _51 corr(X,y) 4-1 iAJ,1 -1=LJVI J-46-4 V(Y)
:411:311
Corr(X, n = P X ,Y = 11V (X) • V(Y)
Cov(X ,Y) (7 — 5)
cr x -cry
ç jL JVI 11% sL:.9=„X, yau...141ff p x .3=14:3. 511 Lidatm...4
• gr"-C'
: 13145-111 LI-42A C.)411-91 0122:1 - •
(1) P x,y = PY,X
. L531-1.b i P xy -1.1-.25111 J-41-2-4 L 1.1AJ
ETV — E(X))(Y — E(Y))]
0
ri
- 162 -
-1- 215 s.),..1.4
Li
(2) P x ,x = 1 , (3) Px ,-x = —1
(4) — 1 pxy 1 .
: cil-t.)411
(4):.k,u.--=.1-L11U , 4.53-4 (3) 42)41) ot-k.)4
(4) CJI-A.)-.6 j 4143.1.9 [-1, 1] 'La 111 - 5 JV
X — E(X) z jt,„,,11 _x;1,11 e J1-11 : = o-x
:ui., Yj X C..a).!;1 1 1 13:3.al DI V(Z) = 1, E(Z) = 0
X — E(X) Y — E(Y) Z , = , 2 =
a X
Cov(Z1 ,Z2) = E(Z1 Z2)— E(Z 1 ) • E(Z2) =
= ERX — E(X)
)(Y — E(Y) )]
a x cr r ERX — E(X))(Y — E(Y))]
, = Px,r CY x ay
V(Z, + Z2) = V(Z1 )+ V(Z2 )+ 2Cov(Z0Z2)
V(Z, +Z2) = 1 +1 + 2Cov(Z1 ,Z2)= 2 + 2Cov(Z1 ,Z2 ) = 2 ± 2p x
V(Z1 + Z2) 0 2 + 2pxy 2pxy —2 pxy —1
V (Z, — Z2) = V (Z1 ) + V (Z2 ) — 2Cov(Z0 Z2 )
V (4— Z2) = +1 — 2Cov(Z„Z2) = 2 — 2Cov(Z0 Z2 ) = 2 —2pxy
V(Z, —Z2) 0 _>2-2p ). 0 =-2p,, —2
2 pxo, 2 = px y 1
C771
- 163 -
-1— c_11...1 215 siL-.
c. 3_ 1 -54 -9 — 1 y
(5) P(aX ±b),(cY ±d) = P X ,Y
fridv p (5)
Cov(aX ±b,cY ± d)
1JV (aX ± b) • V (cY ±d)
E[{(aX ±b)- E(aX ±b)} {(cY ± d)- E(cY ± d)}]
IJV (aX) • V (cY)
E[a{X - E(X)}c{Y - E(Y)}] = acE[{X -E(X)} {Y - E(Y)}]
Va2V(X)-c2V(Y) aclIV (X) • V (Y)
E[{X - E(X)} {Y - E(Y)}] =, = is
(X) • V(Y)
: jt-11 .54,3:11 ;LS ji,13.11 6_5
C.]
Joint Moment Generating Function
-
t+I jAjcsall j Jp X , y SJt1I j3.11 41,
:iI"A1-1-21-.3 M(t I , t2 )
M x t 2 ) = M t 2 ) = geXt1±n2 - a) < t1, t2 <00 5
Li )L>.1 O J.; Ola
LI-1S = P (ax ±b),(cy ±d) =
- 164 -
L.)-3-.1 215 s_.).S..1. asz...4 .37,11 ("_)1-.. ..1
ex"2 f (x, y) x Y
co co .1 ex1"2 f (x, y)dydx
M(t i ,t 2 )= E(e x"n2 (8-5)
LI
J.415:1113 b.. J5 DA)
Jj2 .143911 '..4-5J11-411 ;U111 Li49-
(1) Mx), (0,0) = 1, (2) M x ,y (11 ,0) = M x (t1 ),
(4) ari M(11 12 ) E(X'
ar2m(t,,t2 ) ,_. E(yr2 ) (5) u I, 2 ti =t2=0
arl +1.2 M (t i t2 = E(X 11 Y .2 ) (6) Ot;l at'22 „=,2=0
(3)M (0,t2 )= my (t2),
,515,,t,11 (6) cci
+r2
,X , y ja.-11)
: .4-S.;""11 .32-11.9 (41"4-11 ) Z.."1z-11 — r
c.53,11 cj_C.4:2 J 1 LLI3 csA :k4L,31
j i
( 4, 5, 6) u...1_9.1,31 U J .M(t i ,t2 ) caits 131
- 165 -
+ 3t1t2 E(Y 2)-1- • • • + EE 1+ — •
3! r!
2 (Xt H- Yt2 )r
215 s_)-C3-4
Lb" 2"
=E(X), pr2 =E(Y"), ,r2
= E(XrlYr2 )
:‘,3tas,I.Inj a4j_91sA40.1.4
k=0 k!
M(t1 ,t2 )= E(ex"+Yt2)= (xt1+yt2)k
k=0
= + (Xt, + Yt2 ) + (Xti + Yt2 )2
+ 2!
(Xti + Yt2 )3 + (Xti + Yt 2 ) r +
3! r!
t2) = 1 + t1E(X)+t2E(Y)+El(Xt1+Yt2)2
2!
+(Xti + Yt2 ] + +
(Xti + Yt2 )r ]+ •
3! r!
12 t2 2 = 1+ t i.E(X)+ t 2E(Y)+-i—E(X 2 )+-1E(Y )
2! 2!
t3 3ti2t 2 E(XY)+ E(X 3 )+ 2 E(Y)+ 2 E(X Y)
3! 3! 3!
...e = xti+, E (xti +yt 2 )k
M(tl, t2 )
2t1t2
2!
ri 2 = E (r ) , / Pr, ,r2 = - r2 ej_:),31 (Di Li
JC; Lci
- 166-
(..)-=,- 1 215 _941 OL:u11.11,).44U ..!
: (3-5) JA
LLJ
ri
11
(1) V(X — Y), (2) Cov(2X,3Y),
U-4 (1 -5) ilf5-.1
(3) Pxy (4) P2X,3Y (5) Mx,y (ti '12 )
. E(X)
:
k-,‘ (1 -5)
(1) V(X— Y) = V(X)+ V(Y)— 2Cov(X,Y)
Cov(X,Y)= E(XY)— E(X)E(Y)
=1.75 — (1.8)(1.05) =-0.14
.*. V(X — Y)= 0.96+ 9.0475— 2(-0.14) =10.2875 .
(2) Cov(2X,3Y)=(2)(3)Cov(X,Y)= 6(-0.14)= —0.84 .
Cov(X,Y) —0.14 (3) P X Y = = 0.047 .
a, a,(0.98)(3.01)
(4) P2x ,3y = P x = —0.047 .
(5) M X Y (I-1/ t2 ex1"12f (X7 Y)
= el1-212 f(1,-2)+ e" f (1,0) + e"512 f (1,5)
e311-212 f (3,-2) + e311.f(3,0)+ e3(1+512 f(3,5)
M x y (ii ,12 ) = 0.15e!'-212 +0.25e11 +0.2e11'512
+0.2er-212 +0.05e3`)
- 167 -
215 s_)5.3-4 4-4--N-4
at, ti =t2 E(X)= {0.15et1-2t2 (1) + 0.25ert (1) + 0.2 et'+5 2 (1)
+ 0.2e3t1-2t2 (3) + 0.05e3" (3) + 0.15e3"92 (3)} Jr.()
= 0.15 + 0.25 + 0.2 + 0.6+ 0.15 + 0.45 =1.8 .
c. („1., o_*)lc.1 (Di 1=N E (X) = xf(x) t41.141 4,1.IC. di, LA Jab .3
1-4,3 (4. ,•-•1 (2-5) L.111,1' : (4-5) ci1:14
(1) Cov(X, Y), (2) V(X + (3) p.
-:c.r1,11_4,..s (2 -5) J113 csi t.:141.1 (3A
(1) Cov(X,Y) = E(XY)— E(X)E(Y)
E(XY) =
11, = —
8 0
1 = —[2 8
22 1 xy(x + y)dydx =
r— o o 8 8
y2 Y 3 ,2 _1 x2 x
2 2 .1
o
0
=
jrx2
(x 2 y + xy2 )dydx = o
4 8 _ —+ x—]dx — 2 3
4 — =1.333 3
L —+ —.ho ax -- 2 3 8
x3 8x2 2 2 + ]20 [—+—]
3 (3)2 3 3
Cov(X,Y) = —4 —(-7)(-7) = --1 = —0.028 . 3 6 6 36
F71
Li (2) V(X + Y) = V(X)+V(Y)+ 2Cov(X ,Y)
1 11 —1 20 V (X +Y) = 1 —36 + -T-6+2(-36 )=
36 =0.56
—1
Cov(X,Y) (3) pxy _ = a X Y I11 11 11 a —
1136 36
_1_
iaw(ti,t2 ) E(X)=
iiJ
— 168 —
Li
Li
-1-j-.3.1 D---1215 ,9,11
: L.).4-41-L11 Ci 1•1 (s C..):1.J1-4:1 7 — 5
Cr x = GOV (X Y) = E(XY) — E(X)E(Y) J,L,Lall c4,1 (1)
Cov(X ,Y) = 1 — {V (X) + V (Y) — V (X —Y)}:*A:s"N-a 4 (2) 2
(1) P y = Py,x , (2) Px ,x = 1 , (3)Pxx = —1 : J4 (3)
: (4)
(1) Mx,y (0,0) =1 , (2) M x ,0) = M x (3) M (0, t2) = My (12)
LJL L.s .,1 (5)
LJW Ly, L jt3S1
PL-=..- 1 4 =X : DI...}.2;1^11 1313 ;1-3:1 _92c'L> *A.` Y.'"
cci
(I)
(y) (X, y) L:),4 ;k' —II 41 (i)
ft)
(1) P (X + Y 1) , (2) F(0,3) ,
(4) V(X + Y) , (5) px y
(3) Cov(X, Y)
(6) M x y (ti t2 ) •
1.64jic Gi b aX, bYau4i1. J:).11111 Di L:A_):1(i) (6)
Cov(aX ,bY ) = ab Cov(X,Y)
C171.,
- 169 -
-1- J\--", -.1 215 CJI71..):2-"1.)-au
L,16 x,y -As 31,t,11;k_.41....:sil .;61.) 131 (L,j)
x + y f (x,y)= , 05_x,y<2 .
8
(.5.1.."
(1) f x (x) fy(Y) , (2) F(1,1) , (3) Cov(X, Y)
(4) V(X + (5) Px,y (6) E( xX+ y ) 5 (7) M Xy(ti,t2) •
a AAJI.11
ni Li
Conditional Distribution cs6J2111 t4-1.-3,9:111
: ZA-14 1 — 6
: 3 -.?1f3 ..j.,411'..L% 413 j2,11 j1.4:Lz' .Y1 144 c:)A
P (A I B)= P (AB)
, (1-6) P(B)
J.: ;1,11 L• ji
A = {X = , B = {Y = 5 C = {X , • • •
(1-6) 1.31 (.53
x, y)
P(X = x IY = y).= P(X =Y = y) , (2-6)
: c_cluts3 (2-6)'A-s--) 211 C.)1 LsA f(*)
f I Y) = f (x,Y) f(y)
- 170 -
Conditional Probability Function: .4.7?13.),111 .44. 1-4:1='11 ZI1.511 2 — 6
:
f(x) j1_9.111,3 f(x,y) j;,t,11 L3L41%/1 ;01-) x,y J iAli Dts 13)
X .J4 - LAI] J j l L31-4.`il :61.) f(y)
f (y)> 0 (3-6) f (y)
:uj LD1,1.1:;11 Li
f (x y) = 1 - L
(i) f (x y) 0, V x , y. (ii) f (x I y)dx
-CO
Li
Li
: (1-6) JtiA
( .11:111 di.1,31 4i US c,_fo ( X y) Z.11.1_11 ZILD C.I315 12.1
: : (=1_5311-411
• fX (X) .fy (Y) 4i Jj.1.11 (1)
(lx iU -:).__).J'311 (2)
P({X = 2} U {Y = 4}) (3)
P (X =1117 < 4) (4)
f (x, Y)
Y
2 3 4
x
1 1/12 1/6 0
2 1/6 0 1/3
3 1/12 1/6 0
- 171 -
3.1,1 215 s CJL:L}:51..11,}4,411.1
:
: L:)-..141. 1111 all. J-14 (si fx (x) fy (y) L-11 -5-'11 (1 )
x 1 2 3
f(x) 114 1/2 1/4
Y 2 3 4
f ( y) 1/3 1/3 1/3
f(2,2) =
1/ 6 =1/2 , ,f(22)=
f(2) 1/3
, , f (113) =
f(13) 1/ 6 = 1/ 2 ,
f (3,3) =
/ 6 = 1/ 2 ,
f(3) 1/3
f (2,4) =
/ 3 =1 , f (214)=
f(4) 1/3
f (1,2) 1/12 f (112) == =1/4 xly f (2) 1/3
f (312) = f (3,2)
= 1/12
= 1/ 4 f(2) 1/3
f (2,3) 0 f (213) = = =0
f(3) 1/3
f (1,4) 0 f (114) = = — =0
f(4) 1/3
f (3,4) 0 f (314) = = — =0
f (4) 1/3
: L.L9.141 csiL .. X j -11 4—A14.1 4--31c-9
f (x I A Xty
Y
2 3 4
x
1 1/4 1/2 0
2 1/2 0 1
3 1/4 1/2 0
Li
- 172 -
-1- 3-.1,1 215
L111 (-3_511 \-#4" cr4A (fx(xI Y)) X .).4;=1-43.1 Z.4L;-.4-11 =_A
x 1 2 3
—.-fx (x I 2) 1/4 1/2 1/4
x 1 2 3
f x (x I 3) 1/2 0 1/2 n,
1 .1
Li
r
x 1 2 3
fx (x I 4) 0 1 0
:346 ,-,3_311--11 (3)
P(IX = 21 U {Y = 4}) = P(X = 2) + P(Y = 4)- P[{X = 2}n {Y = 4}]
P({X = 2}U {Y = 4}) = f x (2) + f y (4) - fx,), (2,4)
=(1/2)+(1/3)-(1/3)=1/2 .
yin--11 (4)
P(IX = n < 41) P(IX < 41) =
P(Y < 4)
Li fv,y (1,2) + .ix,y(1,3) = (1/12) + (1 / 6) _ 3
8 f y (2) + I;, (3) (1 / 3) + (1 / 3)
Conditional Distribution Function: : ;k7Jz. t471..3.92 .A-11. 3 — 6
:
F(x I y) _5-.)11-.1 X
F(x P(X = y) (4 — 6)
- 173 -
u- 215 S._)-CL4 1.Y"=21-1
,^1 i
F y) =
y)
f (u y)du k-45 X J 11 t_53 L7L-43
Li
. F(x)..3 F(x,y) F(x jy) L.r=.1 _5.:, (1)
a
F (x I y) = f (x I Y) (2) 0
csAjL&fl (...:.131.S 2-6 dila fl
f(x,y)=2 , 0<x<y<1 . fl
: ciA
(1) f (x) , (2) f (y) , (3) F (x I y) , (4) F (y I x) ,
(5) f (x I y) , (6) f(3; I x) •
:d
1
(1) f (x) = f (x, y)dy = 2.1 Li
dy = 2[y]l: = 2(1—x).
• f x (x) = 2(1—x), 0 < x <1
(2) f (y) = f (x, y)dx 21 dx = 2[x] )01 = 2y . 0 0
fy(y) =2y , O<y<1 .
- 174 -
F(y 1 x y - x 1 - x
1 Y = 1
, y = x
0<x<y<1
4
(3) F(x y) P(X xlY = y)
x f (u y) 1 = f (u I y)du
f , du = 2du = 1 —[u], =& 0 f (y) 2Y
, 0<x<y<1 .
I: 3x<0
Li F (x I y) = 0<x<y<1
, 1 y
(4) F (y I x) = f (v I x)dv = f (x 'v) dv = 1 2d v 1 [v] x f (x) 2(1- x) x 1- x
F(ylx)= y1: , 0<x<y<1 .
JLQfl L L ‘,113LC.,y .4....)'?/ (5)
LAIS b-)1,c.1 Las L:•31•1,tkil
- 175 -
-1- L31_42.1 U:1i 215 "Li-C3-4 C..11-15911 JAZIL
f I =
f(x,y) = 2 = 1 0 < v<
1 AY) 2y y
, 0<y<1 fl ax ax y y
T.1
f(x, y) 2 1 , 0<x<1 .
(6) f I = 1 o<x<1 ay ay 1—x 1—x
0
Conditional Expectation : u12.1411 .a..3Z 4-6 9
v._335 L.4.1 JAI (sic. ais: x,y j.jia..1" I ul.S
Y=y ti U jj g(x) 2.11.11 ‘).
:1141t11 Z97-)61.1(04 1 t_5:'
Eg(x) f (x Iy)
E(g(X) I Y = y) (5 — 6) f g(x)f (x I y)d.x
Fl
f I utri -11 c..)-4 cis LIA CJI
X J.:;1-11
q(x) Z11.111 c..1* Vta.11
g(X) = x L.:31s 131 — 1
E(g(X) = y) = E (X I Y = y) pxy
- 176 -
f (x) 2(1— x) 1— x
11 L_J
-1- i'- U"' 215 k)s3-.
JJ Y ' X J.!;1'31 th.3.-111 .313 ..9
. E(X IY) , E (X) :14.3.4 j.c:13 jya
g(X) = (X mu xiy ) 2 ci:ILS 13) — 2
E(g(X)1Y E[(X 1-1x10 2 Y = (5-x2 iy
4.1 oj,33 Y L:m1A 134 J2.;'. X .J4;13-2.11 cpl.Y2. CD:A-.6.113k9
V(X I n
Vx,,, (X)
: X j i cr5 )..),111 c1341 Di
V (X in = E(X 2 I ,Y) — (E (X 1Y)) 2 .
:D13 g(x). xr cL31.s 13) - 3
E(g(X)IY = y)= E(Xr IY --= y) -a- 11,1
. x .):?;i3.11 r Z.6.)11 d--.S11 ";Jpc5-1:a_A (-3,31.5A3
: (1-4-6)
: x, Y
E[E(X IY)]= E(X)
x (.5_,La. 411 tip Lyii3 ( y L6A) E(x in j..4ia..1.1* _*k j:111
: -I t70.5.3 -¢ 1 X ,Y ülj;JSb LY-L41-1.3
C- E[E(Y X)] = E(Y)
- 177 -
_1- Lit..11 215 OL:u1-11 JA=12,
csi ;1.2...)1d11 c:AL :(:)1.14„,411
• E (X I Y) = xf (x I Y = y)dx —CO
CO
• LHS = E (X I Y = E(X I' Y) f (Y)dy —00
co 00 f' f xf (x Y = y)dxf (y)dy = dx f (y)dy
00_.0 f(y)
= fx f (x, y)dydx = fxf (x)dx = E (X) = RHS .
: (2-4-6)
: C.J1i Ls-11-zz.1 .333 t.+1 X ,Y C.31 J.!;1-11 tS 13 )
V(X)= Ey[V (X 1Y)]+Vy[E(X 1Y)]
V(XIY) c5i=1 1 C14141 : 94 X J.:;1A ce141-8-11 &AA CJI 43'71 131
4-11-41.1 cj L'4+1 X ,Y W-11 (DIE 13 ) JI411-.1 _3 E(X j Y) 5 -1,_A31 tij 1 L:),11,15i
V (17 ) = Ex [v(17 1.70]+vx [E(Y X)]
J.,1.11J1_41z.V1'.1_11.11:Ljal d_513) (3-6) jt.14
f(xly)=1 0<x<y<1 .
(:).4J (....s.t.a.z.ti
- 178 -
-1— c..31-411 u--1215 OJAC.1.14-3 6-3.-;9131,)•"21-1 4-43-4 .
(1) E (X Y = y) , (2) V(X I Y = y)
:
Y (i) E(X IY = y) = xf (x I y)dx =
1 — xdx 1 [x 2 ] y
o' = — ,
Y o 2y 2
0<y<1 .
(2) -.• E(X 2 1Y = y) = f x2 f (x 1 y)dx = —1 1 x 2dx = —[x 3 ]-(; ) = y2 Y 1
3y o 3 Y o
.'. V(X I Y = y)= E(X 2 I Y = y) — ( E(X IY = y))2 2 2
= .Y ( Y )2 , Y , 3 2 12
Y2
1-2
0
, 0<y<1 .
- 179 -
-1- Jus-1 (:)..+?.J
: :t•1 5 -6
Moment Generating Function for Conditional Distributions:
:Jj
ruu L526 f (x I y) 451-L. g511-z-1 &._3,55 X ,Y clit 5 131
(o..1.1.?„3-4 .L.11.5 13)) gLJ &._13:111 e_332.1 sa1_,..411 11.1L11 I' 1
E ext f (x I Y= y)
M (t) E(e xt Y=y
fextf(x 1Y = Adx -00
J,15:111_, ti,-;•,,11 D.4 J5 DA Di LA
: oalsa-11 Liat2t1
(1) M (0) =1 .
ar (2) —ar Maly (t)11,0 = M, (0) = E(Xr 1Y =
91 Ls-1" X j-‘1 r Z.6.)11 _33 Ls-k)2111 JAJ
: (4-6) (.1114
0<x<y<1
D4 J5 1+1. MxlY (t) L_J.11 • .
(1) E(X I = y) , (2) V (X = Y) •
- 180 -
215 I (3
M (t) = E(ext IY = y) =
co
1 Y ext
J ext -1 f (x I y)dx = — f dx = —[— Po'
o yt
eY1 — 1 yt
iii (yt)i. 1
> M XIY( t ) ) yt
1 yt (yt)2 = —[(1+ —+ +...+
(yt)r + • • .) — 1]
yt 1! 2! r!
1 yt (yt)2 (yt)r= —[-
1! 2!+ + • • - +
yt r! + -]
=1+ —yt
+(y02
+ • • + +• - (yt)r
2! 3! (r +1)!
. .
(yt)r tr Yr __L. (r+1)! r+1 r! rb(Til r +1
(1) E(X IY = y) = — 0 < y < 1 2
(2) V(X I Y = y) = E(X 2 Y -= y)— (E(X IY = y))2 2 2
Y ( Y )2 _ Y 3 2 12 y2
V(XIY=y)= 12
. (34) L.11.1.j.
yt
- 181 -
f (x,
-2 2
1 0 0.2
x 2 0.3 0
3 0.1 0.4
_1-jl41 215 sisl. Lk _9;11 •
.)-4:1 6 — 6
f(x I y) L.:4i (1)
:
Zf (x I y) =1
(1) f (x y) 0, Vx,y. f(x I y)dx =1
II) c:3 Y J.:;1-11 L,31-4jji (1-45 ciU (2)
E[E(Y I x)] = E(Y) . LL431 cj i j1.4c,t4 (3)
r3 Er
M x (0) = 1 3
01.1.11 1.A.S Lro, x / y L:JL*1 (5) 3
f(x iy) ZlaS.11 'Al.) (1)
F(x I y=2) (3) , f(x I y=2) (2)
(4) E(XI Y=2) , (5) V(X I Y=2) , (6) M x12(t)
- 182 -
174
-1- Jus..1 u---1215 sic,14 _9,si Ljt:1, ,J
c).4.41111
Gi.L.g2:111 LN:61.41) L:11j,ic.1.411 j`.)11,,.11
Independence of Variables (Stochastic Independence):-
: List...3 1 — 7
j•ji3,,%/1 *A.+ _9!:-11 1_1! Dc..1
123 .1.1 1.11 V.)) ullt;1,.,4 LA.g.:11 Bi A-.4j.cul*..t.. J14
: Ji
(1) P(A n B) = P(A)P(B) , (2) P(A I B) = P(A) , (3) P(B I A) = P(B) .
A {X x} , B
c.,11:11S SA1211 ( .6c; cji
(1) P({X x}n{Y = P({x x}) • P(IY ,
(2) P({x x} 1Y = P(Ix (3) P({Y {x x}) = P(IY yb •
41.4_,
: 2 - 7
ccA, kLLi X , y ulg 9 t..11 /DIJ.;4-;.111 LDLC 1'4
1,61 13) LyJLILL,,.. 1,1.11 Ju F (x, y) , F (x) , F(y)
fl J' -1
— 183 —
(1) F (x, y) = F (x) F (y) , (1-7)
(2) F(x y) = F (x) ,
(3) F(y x) = F (y) .
I fa t7-111 3 — 7
;As :).1.1.4. 411-4,...11 J1,9,111 1- 3-7
:411:th cL,s ,4X I y (1,4_52,231 cjI _1;JS 131
(1) f (x, 32) = f (x) f (y) (2 — 7)
(2) f (x I = f (x) , ( 3 ) f (Y I = f (Y) •
: j411
A {X = x} , B {Y = y} (1)
P(Ix = {y = = P(Ix =
••• f(x,.Y) = f (x) • f(Y) •
Lr1.91"11 .346 _9
LcItill (1-7);IXN,1LII (Di ( LrJr.i.t-11
F (x, y) = F (x) • F (y) (3 2 a a F (x, y) = — F (x) — F (y) axay ax
f (x, y) = f (x) • f(y)
- 184 -
0
t
0
_1_ 215
C37Lz...J
f (x, y) f (x) • f (y) = f (x) = RHS (2) LHS = f (x I y) = f()
= y f()
(„,.,.11...11 45514) , (3)
-: 4)114 2- 3-7
.. ;t41.) 72, aALL,L. x, Y
(1) E(XY)= E(X) • E(Y) , (2) E(X IY) = E(X) , (3) E(Y I X) = E(Y) .
:(:)114.)41
I
LEA utAi..111 ,,A11 45:133 kL
CO CO 00 CO
(1) LHS = E (XY) = xyf (x, y)dydx = xyl (x). f (y)dydx -co--OD -CO--00
00 00 CO CO
f xf (x) yf (y)dydx = xf(x)E(Y)dx = xf(x)dxE(Y) _co _co
= E(X) • E(Y)= RHS. E(XY) = E(X) • E(Y)
. H1.931"31
- 135 -
215 s 1.-NA
CO
(2) LHS = E (X 117 = y) = f xf (x I y)dx -CO
xf(x)dx = E(X)= RHS.
E(X 1Y) = E(X) .
. YJ"---11 .3AJ
(SLait (3)
-: 04111A 6:1.411. 3- 3-7
L y,s2.11 cjsi j4ra.Lu*-,4X , y DI jj1 DIS 13)
(1) Cov(X, Y) = 0 , (2) V(X ± Y) = V(X)+ V(Y) .
:O114.›.1.11
(1) Cov(X ,Y)= E(XY)— E(X)E(Y) ...Cov(X,Y)= E(X)E(Y)— E(X)E(Y)= 0
(2) V(X ± Y) = V(X)+ V(Y)±2Cov(X,Y), .- .V(X ±Y)=V(X)+V(Y)± 2(0)
= V(X)+ V(Y) .
I
El
- 186 -
f (-1,-1)=-1
= 0.063# f(-1)• f1,(-1) = —5 5
= 0.098 . 16 16 16
-1- 215
..c.}4?., ,..9t3 Cji71.59.11 ,102—NA .4
314 j
El
x
f (x, Y)
Y
_1 0 1
X
-1 1/16 3/16 1/16
0 3/16 0 3/16
1 1/16 3/16 1/16
: (1-7) SaA
au2=1 .11 ,t• ...di a 1S 13)
X , Y LI)ui•LiJ A-A,k).) (1)
. Cov(X,Y) (2)
:
LI 11_ (:)-°L SLa.. ° LD:U.:'1^11 S-361."1 e")c -31 ) (-M-4 cps-1 (1)
: JI L113-1.11 (:)1•1?:, .!N)-1 J.5.14 c.)- 4231-i . J73\11211
6 5
fx (-1) = (-1) = —16 = fx (0) = fY (0) —16
5 .f.x (1 ) = fy • = --16
J (-M-Lusii .;L"11J3
f (x )2) = f x (x) • f(y) ,V x, y
os__Ar-1 2,1-11 SeLa
- 187 -
-1- dt.z,..\ 215 s_)SLA C.4)1.11,y.t:tu
f(x,y) 44 JUL. L353 JUIAII
:th7 au.:;1-11 -43 _} —11
f (x,y) = 0 # f x (x) fy(Y) •
: C.J1 Cov(X,Y) (2)
E(Y)= E(X)=(-1)fx(-1)+ (0) _G(°) + (1)f (1)
=(—l)--.5 + (0)-616
+ (1)-516
=O 16
E(XY)=IExyf (x, y)
1 (_1)(_1)_1
(-1)(0)1+ (-1)(1) 16 16 16
+ (0) (-1)-3 + (0)(0) • 0+ (0)(1)3
16 16
+ (1) (-1)-1 + (1)(0) —3 + (1)(1)-1 =0 . 16 16 16
Cov(X,Y)= E(XY)— E(X)E(Y) 0-0 = 0
J-41-9-4 4— 3-7
u4 u<cfil -.- (:):.11791-. X, Y C.g 13 )
PX,Y -0 • 7
- 188 -
-1- L31-4 1 215
Cov(X,Y)= 0
Co v (X ,Y) P X,Y = =0
Cr a X Y
. Yill= 411 .91 b
13_11 1,411 'AS j.:12.4.41 WI 5- 3..7
c.: U•' x y LDLsr.))
M X ,Y(t1) t2) = M X(4) . M Y(t2) •
:ciut,),111
LHS = M x,y(ti,t2)= E(eXII+Y12)
El =J Jexh1t2f( x,y)dydx
_00_00
00 .0
= ext'eY12 (x)f y. (y)dydx
= ext" fx (x)dxf fy(y)dy
M x(ti) • M y (t 2 ) = RHS .
: D1741 DI- 4 X, Y (2-7) J 'LA
- 189 -
r
y -2 5 8
f(y) 0.3 0.5 0.2
II cs11-41="11 t.,L5_9111 —
x 1 2
f(x) 0.7 0.3
Cov( X ,Y)=0 —
f (x, Y)
Y
-2 5 8
x
1 0.21 0.35 0.14
2 0.09 0.15 0.06 ri
: LS S4-111 — 2
Clieg.W.M*11 1.1:11%.A4 .L:4,1_7111 -- I M x t 2 ) (.46,2.=.1 3
L3-%11
f (x, y) f x (x) • fy (Y) L-J31 Ai cis —
(Si:;11 j "14 ko.i VI t.,1_13:111 i4..11.
E(X) = 1(0.7)+2(0.3) = 1.3
E(Y) = (-2)(0.3)+5(0.5) +8(0.2)= 3.5
E (XY) = xyf(x, x y
= (1)(-2)(0.21) + (1)(5)(0.35) + (1)(8)(0.14) + (2)(-2)(0.09) + (2)(5)(0.15) + (2)(8)(0.06) = 4.55
Cov(X, Y) = E(XY)— E(X)- E(Y)
Cov(X ,Y) = 4.55 — (1.3)(3.5) = 4.55 — 4.55 = 0
- 190 -
I L.)
El
M>
r
-1-
. JN-L-HVI latt>3
ej3.31 .41_4 1'LS j-;::‘,11LL4 — 3
LIL),.111 4_9:2 Si 2iL u aa. cci (I)
: L5-11:311
...Mx,y(t1,t2)=E(ex,1+Y12 )
= E(e x1Y12 )= 0.21d1-212 +0.35e6±512 0.14e"812
+0.09e211-212 +0.15e4+512 +0.06e21' +812
Li
)631 Ci.4 ‘.1.11,1„4
i2J M X ,Y( tl' t2) = M X (t1) MY (t2)
mx = (x) X
M (t i ) = 0.7 el' + 0.3e2I1
& M y (t2 )= 0.3e-212 +0.5e512 +0.2e812 .
t1 ) = (0.7e1 + 0.3e2t ' )-(0.3e-D2 + 0.5e5t2 + 0.2e2
= 0.21e11-212 + 0.35e"512 + 0.14e1+812
0.09e211 +212 + 0.15e211-"12 + 0.06e2"812
(1) Lyli3 gcrei
- 191 -
-1- (.1 1-1 215 c.j. j ),..aLl
J-,4121 csic• C..):1..J1-4:1 4-7
-: Jca.i!L all-L-4 X, Y L:J4yt-IlLDIJ.2'11 (:)‘- 13) (1)
(1) f (y x) = f(y) , (2) E(Y X) = E(Y).
: c.)1,21:111 &Al—iy X, Y (2)
x 0 2 3
f(x) 0.25 0.5 0.25
y -1 4 5
f(y) 0.333 0.5 0.167
. Cov( X ,Y)=0 J — 2 . v.i B
13 M X,Y (4, t2)
1.483 :Lea auoa. x, y ci_c..4 (3)
f (x, y) = xye-("Y) , x > 0 , y > 0 •
F(x) , Fy (y) cis -2 :61.1 — I :
, P( X >2 Y) (DA (..$ ‘,1,-z.1 - 4, X ,Y au.:;1-11 -3
• f(xI Y)
cs.te L4+.1:tS i.332.11 .4.35 al • au...X14X, y cis...4.1 (4)
M X,Y(tl,t2)= 1- atl — b
1
t2 + abt1 t2 , t< a —
1 , t2 < —
1
: y_311-11
. X ,Y C.):.U.!41-11 Ji.JciAJA — 2 • Mx (4) MY (t2) JA Js .1?-31 - 1
- 192 -
-
-1- c_31-.11
Distributions of Functions of Random Variables (Transformations):-
e31 O.C1. JC, SiLy1c C_1 ') J13.)
i.1p :it 11.!1-11 3.1.11 blt6 31-4L.Jai.1. Lci „
ZP-11Li 1 -N.11 ;Lull J.D
Discrete Variables : .4.1 —"1 -11 c-l'U+;t:L411 : C2,31 1 — 8
The case of one variable J.71L-ca. .4-11- (.1..9- 11 1-1 — 8
f x (x), x = x,,x2 ,• • • 4,..5-16 443 Y3. 1-411"1 A:111C ."431• X L...g
k.a.493 Z11.) c..}:ciii.L3 y p X L k y g (x )
La) Ls* i;Jts, Cji3 .f y (y) , y = g (x 1 ), g(x2 ),• • • . csA
(I)
one-to-one correspondence ULLY-i X Y g(X)
•
:;d1 LIC Di
y L.511:) y i j Jt94 x
; u.4L y 1Lifl -.0.;c11UUus,.!
fy(y)= P( 7 = Y) = P(g(X)= y)
= P(X = g-1(y)), (y) = x1 ,x2 ,• • • .
= f,(g-1(y)) , g-l (y)= x = x i , x2 ,• • • .
; y 1_41..%)11 Z11, Di 41
f;, (y) = fx (g-1 (y)) , (y) = x = x1 ,x2 ,• • • (1-8)
- 193 -
:Usu.
4111c x S13) -;(1-8) J1.14
. Y =X-3 fy (Y) -1?•_93 f x (x) = , x = 1,2,3,4,5.
Z...111:111 L:_*31 t4L._!1'
Y c,71s:
Y = X-3, x =1,2,3,4,5. = y = —2,--1,0,1,2.
: L.51 J'^-;4 Y Lyaix-4 4 2
...Y=X-3, =X=Y+3.
: 431:111SL1Lsic, (.3---`'4 (1-8) :6--)631 LS4.1.3 — 3
fy(Y)= fx(Y +3) = + 3 15
fy(y)= Y +3 , y = —2,-1,0,1,2. 15
11-11 . 3A3
: L531133i Jja-?I.3 1 -111 CJJ13:3 LD1 LD—C471_9
x 1 2 3 4 5
f x (x) 1/15 2/15 3/15 4/15 5/15
Y -2 -1 0 1 2
f(y) 1/15 2/15 3/15 4/15 5/15
fl
1:1
- 194 -
-1_ 215 •LiC-1.
41.1:111 (‘1.1)
X Y g(X) (Jc.t", Di
y -j J-
" uta,c. xi1,x12,...x. ut, j;i_y3
fy (Y) P (Y = Yi) fx (xi, ) fx (xi2) + • • • + f x (x. , (2-8)
-:(2-8) Jels2
f, (x) x ±1 ,
X = 0,1,2,3,4. crtb 4111C a) i_)i;la X (:)1- 15
f ( y)
Z.4.1:111 cL1_51.-,All
Y _x;1,11 _ 1
Y = (X - 2)2, x = 0,1,2,3,4. y = 4,1,0,1,4.
csk,
2
Y = (X -2)2 , Alf = ±( X — 2),
= (X - 2) , x = 2
(2-8) Zi:36a -3
• fy (y) fx (e (y))
f• y ( y) = f x (2+ j))+ f x (2- jy) , y = 0,1,4.
: L5-11:a-C .;61.-'11 L5-lc- _9.463
- 195 - L-J
1 c_31-6---1 215
y = 0,1,4.
f y (1) = f x (2 +)+ f x (2-J)= f x (3) + f x (1)
4 2 6 = —+— = — ,
15 15 15
f y (4) = f x (2 +)+ f x — = f x (4) + fx (0)
5 1 6 = —+— = — .
15 15 15
: gs)L;31 J3-14-1 a.C3M1 a1.4 LDJUI D-C472_5
x 0 1 2 3 4
f(x) 1/15 2/15 3/15 4/15 5/15
y 0 1 4
f ( y) 3/15 6/15 6/15
-:(3-8) JVIA
fx (x)=1(1
)I-1 , x =1,2,3,••• . Le% iJ.4;11-4. X CJ1S 13)
Y = X 2 Li fy (y)
cz_11_,L&.11 &LI
L.5-1c• Y —
- 196 -
El
r
ff.!
9
9
I Lit -1-1 215
Y = X2 x =1,2,3,". =y=1,4,9,....
Lsic' Y--2
-.• Y = X 2 , = ± (x) = x =x .
LJjlL L5AG J.L.411.1i (1-8) -3
y =1,4,9,—. 4 4
,JAJ
: (4-8) LY-LA
4_11:111 .,1._:)_5:1114.V3Ii.Lc" 1..),;•;„ x
x -2 -1 1 2
f x (x) 0.25 0.25 0.25 0.25
J.:41'\1 iJJ
Y = X 2
. X ,Y — (2) (Y) - (1)
.17(X + Y)•;\ (4) . Cov(X, Y) ;u4s• -(3)
f(y) - (1 )
:
Y t 1 I 1
Y = x =- ,1,2. y = 4,1,1,4.
171
- 197 -
215 L..c.Dc.1522. 1,)-Q1:1
: Y
Y X2 5 ..117 =±x -T = x
: L5 L)-'-' (2-8) Zi°1,11 L.S*' . 3
fy (Y) = fx (g -1 (Y))
y=1,4.
; U1.111 r...19: L5.1c. , jai
=1,4.
=0.25+0.25 =0.5,
.fy (4) = .ix(*NF) (--jt) = /ex (2)+ fx (-2)
=0.25+0.25=0.5 .
y 1 4 :j44.11 cjiall 4.5i US
f(y) 0.5 0.5
LA, us Lt.J.)1 u..u,1 .34 f x ,y(x , y) L4.,3 :L31 j.11 — (2)
Y = X2 ,x = -2,-1,1,2 {X = -2, Y = = fx,y (-2,1) = 0 ,
= -2,Y = {X --= -2} = fx , y (-2,4) = f (-2) = 0.25,
{X = -1,Y = 1} = = -1} = fx , y (-1,1) = f (-1) = 0.25,
{X = -1, Y = 4} = 0 fx,y (-1,4) = 0 ,
{X =1, Y = 1} = {X = f x ,y(1,1) = f (1) = 0.25,
0
- 198 -
-1- 3-4'21- 1215 ji j-4
{X = 1, Y = 4} = 0 fxo, (1,4) = 0 ,
{X = 2, Y = 1} = 0 fxo, (2,1) = 0 ,
{X = 2, Y = 4} = {X = 2} fro, (2,4) = f x (2) = 0.25 .
'N.14" 41 DA 4.113
f,. (-1,1) fx (71) • fy (1) 3 0.25 (0.25) • (0.5)
.jJ. ..U4L"'J X ,Y ü 4-3,Y.I:vbi
4_kilij_3_1?31,:5i itt,1„I 45 41.4:1=V t .,-,c; Di
f (x,Y)
Y f x (x)
1 4
x
-2 0 0.25 0.25
-1 0.25 0 0.25
1 0.25 0 0.25
2 0 0.25 0.25
fy(Y) 0.5 0.5 1
Cov(X, Y
• E (XY ) = xyfx (x, y)
= (0) + (-2)(4)(0.25) + —1(1)(0.25) + (0)
= (1)(1)(0.25) +0+0+ (2)(4)(0.25) =0
(3)
- 199 -
-1- L31-1,
E(X)= >xfx (x)= -2(0.25) + (-1)(0.25)
+ (1)(0.25) + (2)(0.25) = 0
.*. Cov(X, Y) = E(XY)- E(X)E(Y) = 0 - 0 - E(Y) =0 .
X ,Y
Y) LJLJ - (4)
E(X 2 )=Ex2 fx (x)
= 4(0.25) + (1)(0.25) + (1)(0.25) + (4)(0.25) = 2.5
.*. V(X) = E(X 2 ) - (E(X))2
= 2.5 - (0)2 = 2.5 .
E(Y) =yfy(y) = (1)(0.5) + (4)(0.5) = 2.5.
E(Y2 ) = E y 2 fy (y) = (1)2 (0.5) + (4)2(0.5) = 8.5
.*. V(Y) = E(Y2) - (E(Y))2
= 8.5 (2.5)2 =2.25 .
V(X + Y) = V(X) + V(Y)+2Cov(X ,Y)
V(X +Y)= 2.5 + 2.25 + 2(0) = 4.75 .
- 200 -
3
Li
21 5 1,_)-Cla
jadt21:4
:a:L-41A ( .*J:i..j4iLsa* 2-1— 8
The case of two variables:
4.4 JL uJ x1 ,x2
auliall Lica Qx,x, .A3?'• (51' (DtkY" L44-5 fx,,x2 (x1,x2)
nx,x2 1-43-4 = g1 (x1 ,x2 ) , Y2 = g2 (x„ x2 )
nyir2 (yi y2 ) jj (xi ,x2 )
: ci)
W1 ,1412 Cab 6.}.11:1:111 g2
;111<il Z11.. wi(y, y2) = x1 , w2 (y1 , y2 ) = x2
f(Y1,Y2) .= = YI,Y2= .Y2)
= P(g1 (X I ,X 2 )= y, , g 2 (X, , X2 ) = y2 )
= P(X, = 34/1 (Y1 Y2 ), X2 = /412 (YI Y2 )
("W1 (Y1 5 y2 )7 /4)2 (Y1 Y2 )) •
LsIts , y2 au:Li:L*1AB ';kS Di 4i
.fr,,r2 (3'1 Y2) = fx1 ,x2 (i4'i(Yi,Y2),14)2(y1 , y2)) • (3 — 8)
01.9":3114 yi ,y2 ;LS ;k1-,<1i LEA,
:41.131 b:1,1, xi ,x2 fl JL csi
(5-8) (.3t64,
ja,,11 US LEA ,
- 201 -
fx,,x 2 (xi,x2)
.X2
0 1 2 3
X1
0 0.06 0.07 0.11 0.07
1 0.08 0.09 0.12 0.09
2 0.06 0.08 0.10 0.07
Li
Y j2;1,11 c.511/1
Y=X1 +,(2
-1- L.3\--ai 215 siC' cjL.4 )1.11
: 4-A c•-)1 J.:;1-11 : JA-11
X1(w) =10,1,21, X2 (N) = {0,1,2,3} = Y(w)= {0,1,2,3,4,5}
y J.;41,11 f, (y)
f( y) P(Y = .Y) f (0) = P(X1 = 0,X2 =0) = fx1 ,x2 (0,0) = 0.06,
fy (1)= P(Xi = 0, X2 = 1) + P(Xi = 1, X2 =0)
= fxi ,x2 (0,1) + fxi,x2 (1,0) = 0.07 + 0.08 = 0.15,
fy (2) = fx1)1,2 (0,2) + f1,2 (1,1) + fxi,x2 (2,0)
= 0.06 + 0.09 + 0.11= 0.26
fy (3) = fx, ,x2 (0,3) + fxi,x2 (112) + fx1,x2 (2,1)
= 0.07 + 0.12 + 0.08 = 0.27
f (4) = fx1,x2 (1,3) + f1 (2,2) = 0.09 + 0.10 = 0.19,
fy (5) = jfxi ,x2 (2,3) = 0.07 .
Li
— 202 —
c:ha..t... L)--1 215 •i3
41111 J.3.141 y 1 431...:11 43.."
Y 0 1 2 3 4 5
f( y) 0.06 0.15 0.26 0.27 0.19 0.07
. (.1=tg-L'i-41 1-2- 1 - 8
:(1 - 8) .471..)J-1
Ls-tb L4-2" L3.1 "k-11.• L.:q1"" XI , X2 1:))
r "‘-11 '.1 431-3111 (-5-1 fX, (xI) 5 fX, (x2)
:\15614 Y= XI + X2
E]; fr(Y) = .ix,(x i )• .f x,(Y - x i ) , (4-8)
LHS fy (y) = P(Y = y) = P(X, + X2 y)
= P(X2 = Y xi) = fx 2 (Y
f(x) (xi, x2)
fx,(xi) fx2 (x2)
S-7,1 tr::1 LIJ
X1
xJ
fxi (xi ) • fx2 (y - ) = RHS .
. Yill= 431 J9
t'7,1
, — 203 —
P"ol
-1- 215 1-15.91- .1.4z—ea
X0X2 :(6 -8) SIA
(.531P (.5-1=, 2 , 8 -11-x-a:i ay-ks-.1 JJ
Y = X1 + X2
t.As%x , X2 L:i,u0raAll.
21 f = , x1 = 0,1,2,3, • • •
x1!
O x2e-G fX2 (x2 , .X2 = 0,1,2,3,•• -
x2!
: (4-8) AI1,,-11 6.2:1-1:3 (1-8)
fy(Y)=Efx,(x)-fx,(y-x)
: L5 11:11 X1,YX1 D:3.,),,i1Alies: (y)
fx (x1)> 0 x, = 0,1,2,3,••• ,
Cr2 = fX 2 (y — >0 y —x1 = -= x1 =
oy-x, e-e
••• fx, (x2) = f (Y — x1) = (y - x1)!
, x1 = 0,1,2,•••, y.
7
fy(y)=Efx,(xi)• .fx,(Y - x i
= yY ,1e BY-x' e-G y!
x':=0 xl ! (y - x1 )! y!
e-(A+9) Y)tx' 0Y-xi y!
.Y! x,--oxi!(Y x1)!
e-(.1.+19) y Y! 0Y-xl , = 0,12,• • -,y
y! xl:=04(y —x1)!
- 204 -
fl
a
Li
-1- Jul-1 L) J215 bJ .1.4.sch
e-(1-1-0) y y\
ej)-X1 , = 0,12,• • • , y y! x, =o xij
e-(24-0)
+ 60Y , y = 0,12 3 • ••. Y!
(2+ 0)Y e20 _(+) fy(y)= , y = 0,12,3,• • •.
)1!
. +0) " ^1.3 JJ Y J112-431 jj
C.).:rtt C.}...4 -9•41.4 t.-3-4.?-41 &,13:111 c.31 cies-411 c)A
A € tIA cs,91-5 A-46-44 1.:=11.i NaiA 1-4421-4
Continuous Variables ;11,-1-11 L:11 .j4.-.1.41' 1 .14.1tt 2-8
: • -11-11 i.atb (ji " 1 - 8
k31-111 :(2 - 8) .A.71,,j1:1 4.51..9‘,11
c j1.1:1.•4 a G(y) = h(x)dx , (5 - 8) a
: cp-tb G(y) :;61.1114CLA
dy G G ' (y) = Ii(y),
( 6 — 8)
Lqic:Y 091114 .t21313 y G(y) cslc. :;61.) Zilf.,14'
, h ( .‘3\ Lci y L3.41C111
r',77
- 205 -
-1- 215 s j5.3-4 ' 4?,. 4 _94i
:
-.• G(y) = h(x)dx a
y+ Ay y
... G(y + Ay) = f h(x)dx = f h(x)dx + f h(x)dx a a Y
y+ Ay
= G(y)+ I h(x)cbc
y+Ay G(y+ Ay)— G(y)= h(x)dx
y+Ay h(x)dx
. G(y Ay)— G(y) y
• • lim 60, = hm Ay-÷0 Ay
0<9<1 • GI (y) = Ay Py—*0
Ay—*0
i.e GI (y)= h(y) .
J43
4_11.111 jLiic.14 :13 - 8) :t
6(y)
G(y) = h(x)dx , (7 — 8) a
. Ll*?1.1 altki (Monotone) E(y) _9 Calif' J1 S.-q-4 a
e • G (y) = G I (Y) = h(s(Y)) d(y)
(8 - 8) dY dY
dAt-C:In (.53. 11 y -1 ;k:-;11-.1 G(y) sJi-a (.51 L,-1,Y 1316
y JA÷.,6.114 i1aALci Lci
- 206 -
<x2 --> g(x1 ) < g(x2) Vx, 3 Y = X 2 .
<x2
: tes31 Y=g(x) c - 2
Monotonically decreasing function
> g(xl ) > g(x2) Vx, 3 Y = 1
X2
17-7
I J\-z,--1 J215 j- ' * .j.44.4 _911
u41,
dG(y) , dG(y) de(y) (y)= h(s(y))cle(y)
dy dc(y) dy dy
(2-8) u- CJA L1133
cj_9S,J a-,11 031b 43, .aa (Monotone) :1 /431.1 e(y) 'LL1r,"•.
:441- (g-6 ) -4.1‘)W1 1 .6C' 131 ( (4-") C16(Y) < 0 dy
G '(y) = h(e(y)) de ( y )
dy ( 9 - 8)
Ag 41..)-2L.11
1,5)2-13.4: 1-62,1 Y=g(x) ''UL1 pc-Ji — 1
r.)) Monotonically increasing function
r7--)
L.j The case of one variable -:.3 2-2 — 8
cci y -= g(X) CL:31Sic , (x) i_.).4.;_•14 X DIS 13)
:444 y f;,(y)
- 207 -
-1-
.* Fy(Y) P(' Y) Fy (y) = P(g(X) y)
= P (X (y)) = Fx (g-1 (y)) g -1 ( y
i.e. F( y) = -OP
fy (Y) = —dy Fy (Y) = f x (g-1 (Y))
i.e. f( y) = f x (g-1 (A) dx
dy , (1 0 — 8)
(y) dy
ci
: (7 — 8) (.3V-1-4
f x (x) , 0 < x <1 : te, x 1:1) 0
Y 11.atio Z11.)
t411
dx = 1
Z..111:i31
1 dy 3 L514.3
LAIt y _.9a= .1,33 _ 2
1<y<4 .
L33
. Y=3X4-1
dy 3 dx
, 1 1 MY)=1--3 =
-3 1<y<4
— 208 —
215 1,ENA jiit,yau
: (8 - 8) Lru-4
(X) =-- J X2 , 1 xJL c...11:31S 13)
. Y = e-x L,A y J.2;1,11 Z...1.11_171 :013
ii
t;z3 j-N21
: — 1
r77,4 dx -1
dx dy
dye
e'
L_J j241,11 - 2
x=1y=e-l &x=00y-=- 0 0<y<e-1 .
: J. (10 - 8)*;k-Kii31 (3-.16 -3
y = e' - x = ln y -> x - in y
1 1 (-1n y) 2
, 0 < y <
_94".9
- 209 -
-1-
dx dy
L.;
fl
Lk.1.+. _94 (...)171,52.11.),..c1:1
• Fr (Y) = Y) F• y (y) = P(e-x y) = P(- X lny) = P(X -lny)
—111 y 1 = 1- P(X -lny) =1- .1
dx = 1 +[x-1 ]1 in Y 1
=1+[(-1ny) l -1] = (-1ny)-1 , 0 < y < e-1
=-1(-1ny)-2(-1) • ir (y)=—dy 01 ) = —
dy( -1n
1 , 0 < y < e-1 yony)2
. Y...91"11 316 3 1111." 47.1A 121-=1_3-31-4 J16 _9
The case of two variables :C.J4. csi 4.3-1....5 11 3-2 — 8
: (4 — 8) Z.L)u
(x1,x2) L5A (X1 ,X2 )
4.1.4 g1 i g2 (DI
= g1 (x1 ,x2 ) , Y2 = g 2 (x,,x2 )aul." 1 -111
(c.}°_)C• '11 ) J . 1-42r4 = g1 (x1 ,x2 ), Y2 = g 2 (x „ x2 ) —
(Yi , Y2)= w2 (Y i , Y2)= x2
csA 1. real 2
axi axi 2 aX2
aY2 UY1 aY2
fy,,y2 (Yo.Y2)= fx,,x2 (wi(Y0Y2),w2(Yi ,Y2)) •
(11-8)
- 210 -
_911 CiLi. j..C1 -1— LILga.1 215 siS3-.
Li Y = ' = + X2 ,
• fYi (Yi) J .fy, (Y2)
: 5-1 x„ x2
:
f."31ilaill (S)
y2 ciA JOL-3 — 1
y7 L L).430< y1 <00 Di C.)4
1:Atbi X2 = 0 (:)."—C:i .1c-•
J= a(x,,x2 )
ax, ax2 ay, ay, ax, ax2 a(yi,y2)
ax, ax2 axi ay2 ay2
ax2
0y2 aY2 aYi
Jacobian of the transformation (-39.--111 J-41-2-A _91 Dk7.'.-.3_,S1-?- .11--4 ^'":: II
: (X1 ,X2 ) 11-"\ I DIS 13) :
fivl,x2 (x1 , x2 ) = e 2) , 0< xi , x2 < 00
(.Y1, Y2) (YI,Y2) (1 )
- 211 -
ii
T71
Li t
17-
-1- 215
2
-= + X2 , Y2 = X
1 + X2
x, Y1
X2 = - X2 = - )72 (1 - Y2 .
: _ 3
ax, ax 2 , 2 1 = Y2 = Y1 , = I - Y2 - - Y1 '
2 aY1 2
Y2 1 — y2
.Y1 .Y1
= —y1 y2 — y1 (1— y2) Y
=y1
: Lcic, 4 CI 1
fYi ,Y2 (Yi ,Y2) = e—(YIY2+Yi—yiy2) Yi =' 0 < < .
Lj IL, ,1 • - JAJ
gre4 A+f,14411 JI J.111 (y)
- 212 -
f
f
-1— cik-4 1
u..=:,-1215 -9:11c1)171..)111_>"21-1J-"A •
fy,(Y1)=J fy,,y2 (Y,,Y2)dY2 0
= f Yie-Y1 dY2 =Y1e-YI [y2]0 = Yle-Y1 0
i.e .fyi (y1 ). yi e-Y1 , 0 < yi < 0 .
f y,(Y 2) = fy, (Y 1, Y2)dY1 = J y1edy1 = F(2) =1
0 0
i.e f(y2)=1 , 0 < y2 <1 .
' 1.911=1-'331 9 •
(>114... C:"1. ‘11-- 1 — 3-2— 8
fx, (x1) .3 fx, (x2) istb X1 ,X2 c:J1 -
: (11 -8) 1-cowl
ly,,y2 (Y, ,y2)= fx,(w1(yoy2)) - fX2 (w2 (y1 y2)) • VI (1 2 - 8)
fX, (XI) 3 1,1,2 (x2 ) cs-lb 1-611-4-113-) Xi, X2 DIS 13) - 2
Y — Xi + X2 j2;1.,11 c :gkaa.4.26L
fy(Y) f f A-2 (Y xi)dxi (13 — 8)
: (10-8) JA
\ AI -5 /12 fl"") cr"i 1 '1\ JS a?I'L"°. di72 Cik'S 13)
, Y = + X2 J.!;.."11 -er -I A41. •••?.3
:
- 213 -
1 juL.-, .1 sisL.
x2 :;LiI2LC.11
fx, (x1) = Ale xl , 0 , /11 > 0 .
fX2 (X.2) =_- 22e x2
, x2 .0,22 >0.
: LA usl-a,la (13-8) 'us-A-D(3.445
Y = X1 + X2 y— x1 = x2
. 0 x y
rti
El
fy (y) = f x,(x i )• f x2 (y— )dxi
= J.21e' • 22e-22(Y dx„ = J 21e 21x1 • 22e-22Y e A2xidx1
0 0
= /1.12,2e-22Y f e(A2-A1) xldxi (14-8) 0
1.4113 j .1111.Z. .1?._371 1-1%
: 4-4 (14-8) L:)1 •1?3 ,11 # 22 I-. f-1
fy(Y)= 2,A2e- A2Y fe(112-A1)'' dxi o e(,12-10-zi
= 21/12e-112Y [ Fp' 22 —21
2122 [e-illY e-it2y 1]
2221
_ /11 /12e-22Y
21
[e(A'2-21)Y —1]
- 214 -
C.J:is,..4 _9;11 jg.1. 1 J.4.21.1
: L5A (1 4-8) Di ••1? = 22 = r- ;4J.tal
f y (y) = 22e-4 1 dx 0
= 22e-Ay rx1 = /12ye-Ay y > 0. i.e Y G(2,2) .
Lj t-t • -3 -54
OAV-111 13.1c. c*J:3.J1-4:1 3-8
f x,(xi) fx,(x2) A 44 x1 ,x2 uls 13 ! 4-11 L .1
: Ik-F)6-11-4 Y - + X2
fy(Y)= Jfx 1 (xi) • f x2 (Y xi)dxi
-4_5U X2 N(2,3) j X, - N(1,4) DISJ 3i X X2 DIS 13) .2
:
Y = X,X2
. Y = à .fy (y) 4_91 X - N(0,1) ui 3
fx (x) = 2xe'2 , x> 0 A_LS.11 ;1 /4.1L .4
Y= ly(y)
- 215 -
L›..L..1 215 .;_)L-. • .u.4,1„.1 j 1j4.
ii u .5
fy (y) j1 fx (-1) = —3 1
fx (0) = fx (1) =1. 6
-1-
Y = X 2
Y = X2 (1-.1:?•-. f ( y) 431
f x (x) = x 0 ;
(i) Y = X2 , (ii) Y f y (y)
.6
7
.8 f,
x = 1 -8
, —2 x 6 .
Y = X2 4-z•• f y (Y)
e , 1 x — , — a x a . ; ZitfaJLc:ji U:=1.)4 .9
2a
(i) y = X + a 00 , = X + a
2a 1 X — a • "+ MY ) 431
fx (x)=—e-A x= 0,1,2,• • • . ; JL (Di u j .10 x!
Y =3X4... f y (Y) 431
f x (x) = 1
, x = 1,2,3,• • • . :JSi1Jl.) L. _>4 .11
Y = X2 L f y (Y)
- 214 -
-A 6
F-1
_1- jua..1 LJ 215 JA "u44.4 (.:91.3fit JA4U AASsi
*41.31.1i
D-4 LS 1i P(B) = 0.4 , P(A) = 0.3 DI-C C..):1113.-.13:" A, B ut:LWI 11)
P(Ac nBc), P(AnBc ) , An B P(AU B) , P(Ac)
:0.11.1,110:5."11
,P(B) = 0.5, P(A)= 0.5 DI-C agii— A,Bc_..i3tS 13) (1)
. P(A I B) P(AU B) , P(An B)
13) (2) 3 4
t: 1 (I) _ 6
.141.—va
• jitil LsIc. (y)
P(B) .1_91:iP(AuB)= 0.8 _9 P(A)=0.3 ut.0 11) (3)
. A c B_ utys..111 .4 cittotz. BJ A
DA :*kf'.11,. B 3 0 Ai,A21A3 I:)) (4)
:L.,451-411i j_5.141
L.
ri
. J3431 it.$)
.P(Ai I B) 3 P(Ai) P(A113) P(B IA)
A1 0.2 0.35
A2 0.15 0.45
A3 0.35
— 215 —
-1- JL.Z.1 215
:&11)31J1'91
: 4.51111 crilatIll _;1_13:111 j eta ..)4ii4" I
P (X = x) = x 3 x = 0,1,2,3
. Mx (t) L-r."1 (2) . c cLi2 24 .1.?•31 (1)
=di t j5:3. jtit ) A (;) j1.24.? k.9 A j.4,14.14c. (51)
1-4,62" 41 cs- cJ-iSJI-11) 4-?3_3:1. DILI-4149 U'":111 6' (Da C14,1MA (:.)15J1-"11 ( .131-5' III
‘-1. lit-41*P -11a c.31". X 42,1. 31 01-5.3 LJ2J L5-3C- 0.6, 0.7 j-ib
. A-La J:4-4 Ls-1) 44.4_9:111 4ilj 22.)_9-411
j.!;1.11 :4451_5111 t._19:111 .x ;111:y11.41.," :ili<11 0
31ik4 c L i1 A..11. 4 -E
f(y) ={cy2 , —1<y<1 0 , otherwise
F(y) (2) . c LI1I1 zo.-A .14 (1)
Li4A1_:%.11 jrp.41
Mx (t) = -4.7 (et + 2)3 ("as X c..5L‘132Lzi1 j.;;IA .3%11 S 1j.4Il A.11.111 13)
.P(X >1) L.11-4:111
: jj.43:3
1.j...1.;a..* X cji.S 13) (2)
Mx (t) = 1(2 + e' )5 35
;LIU LFiaSI (E) . A3,43.3 AIL.41.4 () .X (1)
Mx (t) = : Ls.r4 X L5.Sli_aa j.!;",.1 e_33.11 c.Y.11.S 131 (3)
P(X=3) JS3 A41. l.z.41 II 31 AiU5.11 J13.1 y:61(4)
(1)1° v4
_ (3et +1) io,
CJ:p...1
(1)
(I) Mx(t)= e-2(1-et) (ii) Mx(t)=
Li :
El
- 216 -
Li
215 siL-.
Pt (iii) Mx (t) =
4 - 3
0A1.1L.11 j1,:54.411
: cr1111.11‘.3-c-m-.1:A.4.11 45:11)1 K,-11 X crli.i-'12-11 J.:" ^11 d-s II! (1)
u.
.X c..1-11.542-1! J.:" AlttiJS-111 4.31 0)
. X cril..92,4jit -.9)2-11.i9-431.A-11-111 4.31
: ai X ..)4;6..B c._932-11 S-113.4 k11.) Mx(t) 13) (2)
i) E(X) = M'x (0) , ii) c4 = 1\47< (0) - {Mix (0)} 2
0:..):aL, H J.341 c:31.).4 .ac• .2 X j.;;1,11 I** :k531.4 c.:4.° (3)
L$ 4_91 iT j_96.); JUL,'EQL H ayil;JL
Mx (t), 2 f-tx 4.4.1LL,31.3 f(x) 0 •
My (t) P(X > 3) .3 P(X 4) F(X)
Y= 3X-2
.A, D. %so cik u—A,t c_1431 v4 (4)
5 cci
as-5.7)-4 J UI,.1 ,T(Js'l tslc• L_:241 24) Jai 0
(.31,11,.1
4”1 k11.11 0 c (:),31.g 11. TLI.10 t..9.51A1 ..17-11 0
S. J4 _5110-4 1k35,-4 Ls:aj (5)
. ,,:z1S1 :
— 217 —
215s_)C-14 CL-t4-, jt.,11 ..1
j.i.a.43 pi iate LA.LiJ24 ..) A X c.):._)4 (y)
:10,711 b cli17 Z• SiLll
• X -):!;1'11 11" Y1C1 — 1
4.4_93.3 -b_11.3 4iI ARA j • - 2
M ,(t)= —1 et +-2 e2z +-5 e51 :X 4-51..yta _3_3"..-111,a1J-411 A-Lil 44 (6) 8 8 8
i.33.231,11_3 411 Z11.111 2 .x v124- 11(11 .7k11,1
. Y=3X-2 3 . X
:A.....IL:111 td,-.111:k11.)41,3.:313.2,'ic. 1 LA 11) (7)
0 ,x<-1.
x+1 1.x<2 12
x+4b ,2_<_x<5
4 1 ,5<x
.P(X > 0) (iii) X 4:11 J2.31iLi i:61f611::11U (ii) .b A-4.4 (i)
M y (t) = i tr 7=0 3r
Y J.2;-;-1 _3_3,31 J.4.11 :L11.111 LLVA (8)
.Y J.;;1,11 tisth plr = E(Yr ) e jocil j_c.111
: Z.0:111 .4)19-123'jjiiI CY2 JS3 113113. P1-6:=1A (9)
(43... AL csit (L,_1) ? H el:111 (5_1Q ce_.) 0)
.q (.51,
F(x)
El
- 218 -
jt.....L,1
04.1L.211 (31:9A-41
P(A) = 0.45, P(B)= 0.35, P(A I B) = 0.57 13) (1)
eya
(i) P(AU
A u P(Bc
B) , (ii)
LI-C
P(B I A) , (iii) P(B'
P(A) = 0.4,
D111.41-- Ai
IA) .
P(B) = 0.5, ul-C 13)
B ü0.1.524 -1 :(B_..9
7' IA)
P(B I A) A c B l ji Li-
J L5-1c- X -1,11 F (x) Ub c.:a215 1-4
;:411:111Say-.31
0 x<0
F (x) 1
0 x<4
1 , 4 < x
(i) f (x) , (ii) E(X) , (iii) o-2 c.$5 (I)
;:k411:111 yuial (y)
(i) P(1 <X 3), (ii) P(X > 5), (iii) P(X =1)
:X te .1113 ,t...11 _93.11 11.11.90111 WI 4;11 Liii M x(t)= 0. + 0.3e' + 0.4e2' + 0.1e5'
(i) f (x) , (ii) F(x) . c.J 5 Y ,A=3
. Y=2X- 1
Z11.)u x cyl_C1-.)) (5)
fx (x) = P(X = x) 8 , x = 1,2,3.
c 0)
•j.!;•;.111:1,s3 F( x )
• M(t) (E)
Y L51,1_5.t.11 M y (0 ej_3211JA1ta_111 (1)
LLC
(2)
(3)
(4)
— 219 —
-1-
215 CA.1..5311J 1-1
7:4.444:C111 .A.c._;,:64.11
:c1P1 0:94A•11
1.2.:11...)4i'a4- L:jts 13) (1)
f(x)=1
P(X 0.5) (iii) . 3....)2-11 ;45-‘31:4-11-111 (ii) ..).L=t11 4.33z- r (i)
f(y). —1 , y > 0 tAl y cjt.s C (y) 4
. Y J.:41-11 ei 35 A-11.1i C.)4. _95 YL-1" YilL^U
:01111 J1)0-.411
/3(a + r,b) 1 '4 t- 3.1,I 1 0) Pr/ fl(a,b)
;1..41Um-1,1 klitiS Alb y t..41_92,c• . ALI LILS 13) H
f (y) = 20y3 (1— y) ,o y 1
: 7A-4. Li
(i) E (Y ) , (ii) V (Y ) , (iii) p 51 .
:e.:1112111 (31:5,-4
:Ls:1"-D -.(.321-5 X, Y au211-1111-si."t-11 A-11-111 C.J1 (I)
M xx (0,0) =1 . LJ
- 220 - I,
L_J
Li
fl
( .1
1 Jul.-1 215 jtsi j.2.11,}4.4A.:i AA-1.A
Z-C.)1.'"11 N.:).31 -4 33 t11 - (1):4J X, Y D-C41. (Y)
1 M X,Y(tl ,t2) —
16 (1+ 2eh1 + e12 ) 2 —co<t1 t2
< 00
f (x, y)
Y
2 3 4
x
1 0.06 0.15 0.09
2 0.14 0.35 0.21
:)( j,t-11 _y4;1.11 N3,11 J.4 L31_9.) 4a1 (1)
(H) Mx (t) = t e
e'-1
Z.114 D.4
X J.:;1A C.3;?\-.1.3_3 . Y=X-2 JLA.0 e)13 1,11,3..11
Z-41:111 3,3—.2.11131_43.4 .;;14 X 1:1) (2)
M x (t) = e( (1+21)
?k41,9 ce;'1!.)114-21,9:1;164 1-49 x ii 1-4
4-11,9:1 1-49 Y = 2X+1
.P(X > a) = 0.5C.3i ."44z4, a ;A-4:1j
- 221 -
Li
ET
mx (ti) , my (t2 ) (DAJ ..)t4 : Lr1.5
et.41
412 45;,t,A 1.61X, y
A4 : i-r1.91"11
(1) E(XY) , (2) V(X + Y) , (3) Pxy .
(4)M x (t1 ,t 2 ), (5)M x (t1 )
J1:9,4111
1 (..J1-ALLI Lr.-1215 C 1..y.41L1
L) 4=1215,),514 j .44-1,4611
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: 4f 4S jj 4133 ;k-11 •,) uyc uJ X, Y V--C41 (Y)
M xx (to t2 ) = 1 16 (1+ 2et' + 6.4 )2 — (X) < t2 < CO
• Mx CO My (t2) D-4 :
j.s.au
2
rj
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1 215 Inz..4
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Y
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F (X) =F x (X GO) .
: LA -1: L:Li-a-Nli (1) Fx (x) , (2) Fy (y) .
? 3)vsau,A X,Y 34 -2
4?) :&,,A.L.:- .•)11 1..4 4-1 X iptc II) (1) — (E)
I
fx (x) = r(n) x n
'1' ,x>0,2>0 ,n>0.
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Mx (t) = (2, t )" • -: L.L.1-11
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1 — 1—t2 , tp t2 <1
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0.20 0.20 0.10 1 X
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tt-zli)1LJ
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j1:3.4.411
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:
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(5) Lir (11 4)
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. c 11:4-1 F(x) SU:111 t?..15:311 :1-11•1,,i-,%1 (2)
. m.(t) N37.11 S..11.3.4.11 y.u.t.1 (3)
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2
(4) M x (t)= ez . (5 ) M x (t) = (1-4 )10 (3 e ' ± 1)1°
f.71 Li
r
- 230 -
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M(t) (2) , f(3) ‘7"4-%1 (1)
mx,y(ti,t2) 32-1 4.3:i *A-IL 1 -el c.J-14f11.3,1c. j x,Yu—c:13. (s)
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, tl , 1.2
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-:
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X,Y iJ :tt-.11)1 (,)1."-11
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1 0.15 0.25 0.2 ' fX1Y (11 0) (2)
3 c 0.05 0.15 FAT, (11 5) P(X Y = 5) (3)
- 231 -
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f(x 1 , x2)
(yi ,y2) t1 :k.1L (1)
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fl
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a
: c3.0 JI:Y-411
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X -1 0 1 3
f(x) 0.1 0.3 0.4 0.2
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.
M(t) e_13.-1,1143-4.11 JL 1I ,T,LA.3 (2)
. Y=2X+1 c yJ.:;1-11 My (t) (3)
JL-4111 L5-11)1 ..5.11 X cp-slyIa i 3) (1) - '71
X Li31 9 1.11 (:),J14111.5
=-- E(Xr )= r!
[.1
71
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f(3) LJ(Y)
mx (0 yai (i)
(.31,:9J-4111
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• • • • •
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rt1_31-N-411 'L JA Jj L.14.11 c .v‘S JL (E)
J.5.1 (341.1.311 j.c.) ( .4.1.1511 ) :6111..C11 1L11.1 -
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(2) MA (t) = (1— 2t) 8 .
1 2
(3) x (t) = 2 M • (4) M (t) = e 2
m(tI , .;k..J1 — \1 4_35 a) 1 ^E \ X,11 (:) .7'1, (I)
= MA, (t, )
:41° X,Y 1L› 3-411. ""'"M c-_53.3\ 4_9•3ki Li ( 5)
M (ti ,t2 ) = 1
— 09 < 1, t2 <XD. t + ti t2
— 233 —
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Y
f(x,y) -3 2 4
1 0.15 0.25 0.2
x 2 0.2 0.05 0.15
—
. fAly (1 I 4) (1)
Fxv (11 2) = P(X I y = 2) (2)
: A:11.1.4 1 ULS L-111.1 t4isa N195 &Alt-701,9-1a 1 X CA (i)
f (x) = K x2 e--2- , x o.
V(X) a41-f4, E(X) J 11 (2) . K *A4 (1) :4.4
L.FA A.11.51,1:11_Ac.1.)fitaA* x otS ()
, x+1 .fx lx) = 15 ,
x = 0,1,2,3,4.
. Y = (X — 2)2 L. f(y)
- 234 -
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1 fx(x) = —2a ,
—a<x<a
y f(y) -• . x — a
31:5.4.4.11
ii
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3 — 2x
iY' ci...u..!;12 431 cLiA2c. (1)
:u1 X, Y
Lib
fx(x)= 3 — 2x
, 0 < x <1 2
(i) fx,y (x, , 0 y 1, 0 x
(ii) E(Y I x)
J13.111 (x1 ,x2 ) -.- --11 (Dt-c (y)
1 f(x1)=—e 2 ,
2
fX1 ,X2 (X1 , x2)
--xi x, 0, f„(x 2 )= —2 e 2 , X2 - > 0
: X„X 2 CJI
D.. LISLi ø x1 ,x2 I c:y E-S -13:3113 LJL J\"I J jt4:144 (E )
(i) y2), fy,(Yi)
1 —X2 ) 1 2 = X2 2
- 235 -
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fr1 C.,171.)04.)...411:1 14.14
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0 ,x2 co
XI Y2 = + X2
a.
(Yi , Y2 ) 4, 11 :y1.1_4:1-sV1 : 61LC.11 ZI14 (1)
= +X2
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&;11-4-4111 j1:54-411
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1 M X,Y(41t2)= (1- t1 )(1.- t2)
(i)M x (ti ) , (ii) My (t2) . L.:}4 13-C ‘7-11-w- (1)
• (-74"11 X, Y C.J1. 1-11 34(2)
_ :i.11.1 4.1 (xy) 441.11 11 (y)
f (x,y)= x(1+ 3y2)
,0<x<2, 0<y<1 4
—: LS
(1) fy(x), (2) fy(Y) , (3) fxiy( x , (4) E(X Y) (5) ETE(X Y-)1
:t114,4'11 c.:44:111
- 236 -
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. A c B (3) . B _9 A (2) . DULL,. Bj A(1)
L_ 4 X L,i13,1•0.11 j.!;1.11 DIS 13) - 2
fx(x)=P(X=x)= , x =1,2,3.
• M. CO c-.33.-111,-11_9-4111"-Lli (Lrl) F(x) 33i Lh (y)
. Y=3X+1 Y c;llyt-11 J.:;*"" M y (t) S.113-411 ;;LMI (E) f(y)
:ijU& X
: E(X) 445-; X
4 Fl (i) M( t) = e-2(1-er , (ii) Mx (t) =
7e- )2 , ( iii) M x (t) = (-
4-4
)2 ,
I 3 t
(iv) M x (t) = (1- 20-6 , (v) M x (t) = e32(2.
M x ,0) = M x Lill Z41 1-,-;,11 c.:4;\ - 4
x , y au+;:i..11Zcj; ,11 4s3 ca-ks
t2 +12 M x (t t2 ) = e' CO < t1 t2 < 00
(ii) my (t2
(1) -:
• C' >1-9 L.J.-')-4 X, Y DU,I;L.1-11 LJA
- 3
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Y
f(x,y) 1 2 3 4
0 0.1 0 0.2 0.1
x 2 0 0.2 0.1 0.1
5 0.1 0.1 0 0
(1) E(XY) , (2) Cov(X ,Y) , fl Li
(3) Cov(2X,3Y), (4)Pxy
(5) Mx,y (t1,t2) (6)./x1y (0 I 3)
(7) Fxv (2 1 4) = P(X 21 y =4)
;\ (X, Y) (-1'6 (8)
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_ -.4411:111 ;WO Z11.1 4.1 (XM 4i13,12.11 4.?.:1.0.1' 1 - 6
f(x,y)=8—x—y
32 ,0<x<4, 1<y<3
-:
(1) fx(x), (2) fy(Y), (3) fx ly(x I .31) (4)E(Y) (5)E[E(Y I X)]
_ x 4:11 J.1221 1....1.a.V I :6=1 :t.11.1 - 7
fx(x)= ,x 0
Y J.2;1.11 :Li
c4-111.9 L9 Zujil e-4u)
- 238 -
