Chapter 3-2 Discrete Random Variables 主講人 : 虞台文. Content Functions of a Single Discrete Random Variable Discrete Random Vectors Independent of Random

Chapter 3-2Discrete Random Variables

主講人 :虞台文

Content Functions of a Single Discrete Random Variable Discrete Random Vectors Independent of Random Variables Multinomial Distributions Sums of Independent Variables Generating

Functions Functions of Multiple Random Variables

Functions of a Single Discrete Random Variable


計程車司機的心聲這傢伙上車後會要跑幾公里 (X)?這傢伙上車後會要跑幾公里 (X)?

X 為一隨機變數


X 為一隨機變數這傢伙上車後我可以從他口袋掏多少錢 (Y)?這傢伙上車後我可以從他口袋掏多少錢 (Y)?

Y 亦為一隨機變數

Y = g(X) 隨機變數之函式亦為隨機變數。


X 為一隨機變數這傢伙上車後我可以從他口袋掏多少錢 (Y)?這傢伙上車後我可以從他口袋掏多少錢 (Y)?

Y 亦為一隨機變數

Y = g(X) 若 pX(x) 已知， pY(y)=?

The Problem

Y = g(X) and pX(x) is available.

) ?(Yp y ( )P Y y

( )P g X y

( )( )

( )x I Xg x y

P X x

( )

( )

( )Xx I Xg x y

p x

Example 17

這瓶十元

這瓶只要五元福氣啦 !!!

Example 17

這瓶十元


Example 17

1 ?~X1 ?~X

2 ?~X2 ?~X

1) ?(YP y

1) ?(YP y

2) ?(YP y

2) ?(YP y

這瓶十元


Example 17

1 ?~X1 ?~X

2 ?~X2 ?~X

1) ?(YP y

1) ?(YP y

2) ?(YP y

2) ?(YP y

1 ~ ( , )X B n p1 ~ ( , )X B n p

2 ~ ( )X G p2 ~ ( )X G p

1 ~ ( , )X B n p

2 ~ ( )X G p1 110 5Y n X

2 210 5Y X 1

) ?(YP y

2) ?(YP y

Example 18

1 ~ ( , )X B n p

2 ~ ( )X G p1 110 5Y n X

2 210 5Y X 1

) ?(YP y

2) ?(YP y

n=10, p=0.2.

Example 18

1 ~ ( , )X B n p

2 ~ ( )X G p1 110 5Y n X

2 210 5Y X 1

) ?(YP y

2) ?(YP y

1

1010( ) 0.2 0.8 , 0,1, ,10x x

Xp x xx

n=10, p=0.2.

1( ) ?I Y {50,55, ,100}

1 1( ) ( )Yp y P Y y 1100 5P X y 1

100

5

yP X

1

100

5X

yp

100 10010

5 5

100.2 0.8100

5

y y

y

100 50

5 5

100.2 0.8100

5

y y

y

50,55, ,100y

Example 18

1 ~ ( , )X B n p

2 ~ ( )X G p1 110 5Y n X

2 210 5Y X 1

) ?(YP y

2) ?(YP y

2

1( ) 0.8 0.2, 1,2,xXp x x

n=10, p=0.2.

2( ) ?I Y {5,15,25, }

2 2( ) ( )Yp y P Y y 210 5P X y 2

5

10

yP X

2

5

10X

yp

51

100.8 0.2y

5

100.8 0.2y

5,15,25,y

Example 18 n=10, p=0.2.

Pay 100$, #bottles (X3) obtained?

Example 18 n=10, p=0.2.

Pay 100$, #bottles (X3) obtained?

Let Y (X3) denote #lucky bottles obtained.

3100 10( ) 5X Y Y

3( ) ?I X {10,11, , 20}

3

100 5

10

YX

3~ ( ,0.2)Y B X

3 3( ) ( )Xp x P X x 100 5

10

YP x

10 100

5

xP Y

2 20 200.2 0.82 20

x xx

x

2 20P Y x

10,11, , 20.x

Discrete Random Vectors


Definition Random Vectors

A discrete r-dimensional random vector X is a function

X: Rr

with a finite or countable infinite image of {x1, x2, …}.

Example 19

Example 191

1 ?( ) X (7,3,0)

Example 19

2 ?( ) X (12,0,1)

2

Definition Joint Pmf

Let random vector X = (X1, X2, …, Xr). The

joint pmf (jpmf) for X is defined as

pX(x) = P(X1 = x1, X2 = x2, … , Xr = xr),

where x = (x1, x2, … , xr).

Example 20

There are three cards numbered 1, 2 and 3. Randomly draw two cards among them without replacement. Let X, Y represent the number of the 1st and 2nd card, respectively. Find the jpmf of X, Y.


1 16 6

1 16 6

1 16 6

1 2 3

1 0

2 0

3 0

XY

Example 20



1 16 6

1 16 6

1 16 6

1 2 3

1 0

2 0

3 0

XY

16

,

, 1, 2,3,( , )

0 otherwiseX Y

x y x yp x y

Properties of Jpmf's

1. p(x) 0, x Rr;

2. {x | p(x) 0} is a finite or countably infinite subset of Rr;

3. ( )

( ) 1i

iI

p

x X

x

Definition Marginal Probability Mass Functions

Let X = (X1, …, Xi , …, Xr) be an r-dimensional ra

ndom vectors. The ith marginal probability mas

s function defined by( )iXp x

( ) , ,i iX x P xp X

Example 21

Find pX(x) and pY (y) of Example 20.Find pX(x) and pY (y) of Example 20.

1 1 16 6 3

1 1 16 6 3

1 1 16 6 3

1 1 13 3 3

1 2 3 ( )

1 0

2 0

3 0

( ) 1

X

Y

p x

p y

X Y 16

,

, 1, 2,3,( , )

0 otherwiseX Y

x y x yp x y

Example 21

Find pX(x) and pY (y) of Example 20.Find pX(x) and pY (y) of Example 20.

1 1 16 6 3

1 1 16 6 3

1 1 16 6 3

1 1 13 3 3

1 2 3 ( )

1 0

2 0

3 0

( ) 1

X

Y

p x

p y

X Y 16

,

, 1, 2,3,( , )

0 otherwiseX Y

x y x yp x y

13 1,2,3

( )0 otherwiseX

xp x

13 1,2,3

( )0 otherwiseY

yp y

Example 22

4X = #Y = #

1. pX,Y(x, y) = ?2. pX (x) = ? pY (y) = ?3. p(X < 3)= ?4. p(X + Y < 4)= ?

pX,Y(x, y)

Example 22

4X = #Y = #


pX,Y(x, y)

Example 22

4X = #Y = #


,

3 5 2

4( , )

10

4

X Y

x y x yp x y

pX,Y(x, y)

Example 22

4X = #Y = #


pX,Y(x, y)

Example 22

4X = #Y = #


( 3) 0.1666 0.5 0.3001 0.9667P X

pX,Y(x, y)

Example 22

4X = #Y = #


( 4) 1 0.0238 0.1429 0.1429 0.0238 0.6666P X Y

Independent Random Variables


Definition

Let X1, X2, …, Xr be r discrete random variables having

densities , respectively. These random variables are said to be mutually independent if th

eir jpdf p(x1, x2, …, xr) satisfies

1 2( ), ( ), , ( )

rX X Xp x p x p x

1 21 2 1 2 1 2( , , , ) ( ) ( ) ( ) , , ,rr X X X r rp x x x p x p x p x x x x

Example 23

Tossing two dice, let X, Y represent the face values of the 1st and 2nd dice, respectively.1. pX,Y (x, y) = ?.

2. Are X, Y independent?

Tossing two dice, let X, Y represent the face values of the 1st and 2nd dice, respectively.1. pX,Y (x, y) = ?.

2. Are X, Y independent?

Example 23

,

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 16 6 6 6 6 6

( , ) 1 2 3 4 5 6 ( )

1

2

3

4

5

6

( ) 1

X Y X

Y

Y

p x y p x

X

p y

Tossing two dice, let X, Y represent the f ace values of the 1st and 2nd dice, respectively.1. pX,Y (x, y) = ?.2. Are X, Y independent?

Tossing two dice, let X, Y represent the f ace values of the 1st and 2nd dice, respectively.1. pX,Y (x, y) = ?.2. Are X, Y independent?

,

1( , ) , , {1, ,6}

36X Yp x y x y

1( ) , {1, ,6}

6Xp x x

1( ) , {1, ,6}

6Yp y y

, ( , ) ( ) ( ), ,X Y X Yp x y p x p x x y

X Y

Fact

X Y( , ) ( ) ( )P X x Y y P X x P Y y

( , ) ( ) ( )P X x Y y P X x P Y y

( , ) ( ) ( )P a X b Y c P a X b P Y c

?

?

( , ) ( ) ( ), ,P X A Y B P X A P Y B A B R ?

Fact

X Y( , ) ( , )

x A y B

P X A Y B P X x Y y

X Y X Y( , ) ( ) ( )P X x Y y P X x P Y y

( ) ( )x A y B

P X x P Y y

( ) ( )

x A y B

P X x P Y y

( ) ( )P X A P Y B

Fact

X Y( , ) ( ) ( )P X x Y y P X x P Y y

( , ) ( ) ( )P X x Y y P X x P Y y

( , ) ( ) ( )P a X b Y c P a X b P Y c

( , ) ( ) ( ), ,P X A Y B P X A P Y B A B R

Example 24

Consider Example 23. Find P(X 2, Y 4).Consider Example 23. Find P(X 2, Y 4).

,

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 1 136 36 36 36 36 36 6

1 1 1 1 1 16 6 6 6 6 6

( , ) 1 2 3 4 5 6 ( )

1

2

3

4

5

6

( ) 1

X Y X

Y

Y

p x y p x

X

p y

,

1( , ) , , {1, ,6}

36X Yp x y x y

1( ) , {1, ,6}

6Xp x x

1( ) , {1, ,6}

6Yp y y

, ( , ) ( ) ( ), ,X Y X Yp x y p x p x x y

X Y

( 2, 4) ( 2) ( 4)P X Y P X P Y

2 4

6 6

8

36

8

36

Example 24

Example 24

X Y , ( , ) ( ) ( )X Y X Yp x y p x p y

(1 ) (1 )x yp p p p 2 (1 ) , , 0,1,2,x yp p x y

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

Z1 有何意義 ?

0 0x y y x y z

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

1 1( ) ( )Zp z P Z z

min( , )X zYP

min( , )X Y z

1( ) ?I Z {0,1,2, }

0 0x y x y x z

z y x z z x y z

x y x z y x y z

, ,1

( , ) ( , )X Y X Yy z x z

p z y p x z

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

1 1( ) ( )Zp z P Z z

min( , )X zYP

1( ) ?I Z {0,1,2, }

, ,1

( , ) ( , )X Y X Yy z x z

p z y p x z

2 2

1

(1 ) (1 )z y x z

y z x z

p p p p

2 2 2 2 1

0 0

(1 ) (1 )z y z x

y x

p p p p

2 2 2 2 1

0 0

(1 ) (1 ) (1 ) (1 )z y z x

y x

p p p p p p

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

1 1( ) ( )Zp z P Z z

1( ) ?I Z {0,1,2, }

2 2 2 2 1

0 0

(1 ) (1 ) (1 ) (1 )z y z x

y x

p p p p p p

1

1 (1 )p 1

p

1

1 (1 )p 1

p

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

1 1( ) ( )Zp z P Z z

1( ) ?I Z {0,1,2, }

2 2 2 2 1

0 0

(1 ) (1 ) (1 ) (1 )z y z x

y x

p p p p p p

2 2 1(1 ) (1 )z zp p p p 2(1 ) [1 (1 )]zp p p

2 2(1 ) (2 )z

p p p 2 21 (2 ) (2 )

zp p p p

(1 )zp p p’ p’

0,1,2,z

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

1 1( ) ( )Zp z P Z z

1( ) ?I Z {0,1,2, }

(1 )zp p 0,1,2,z

2 2p p p 其中

~ ( )

~ ( )

X G p

Y G p

X Y

21 min , ~ (2 )Z X Y G p p

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

21 min , ~ (2 )Z X Y G p p

Fact:cdf

( ) (1 ) , 0,1,yYp y p p y pmf

1

0 0( )

1 (1 ) 0Y y

yF y

p y

~ ( )Y G p

121( ) 1 (1 2 ) , 0zP Z z p p z

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

( )P Y X 0x

y x

, ( , )X Yp x y 2

0

(1 )x y

x y x

p p

2 2

0 0

(1 ) x y

x y

p p

2 2

0 0

(1 ) (1 )x y

x y

p p p

2

0

(1 )x

x

p p

21 (1 )

p

p

1

2 p

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

2 2( ) ( )Zp z P Z z

YP X z

2( ) ?I Z {0,1,2, }

,0

( , )z

X Yx

p x z x

2

0

(1 )z

z

x

p p

2( 1) (1 )zz p p 0,1,2,z

Example 24

2, ( , ) (1 ) , , 0,1,2,x y

X Yp x y p p x y

2 2( ) ( )Zp z P Z z

2( ) ?I Z {0,1,2, }

2( 1) (1 )zz p p 0,1,2,z

( | )P Y y X Y z ( and )

( )

P Y y X Y z

P X Y z

( , )

( )

P X z y Y y

P X Y z

2

2

(1 )

( 1) (1 )

z

z

p p

z p p

1

1z

0,1,2,z 0,1,2, ,y z

Multinomial Distributions


Generalized Bernoulli Trials

A sequence of n independent trials. Each trial has r distinct outcomes with

probabilities p1, p2, …, pr such that

1

1r

ii

p


?( )p X n


probabilities p1, p2, …, pr such that1

1r

iip

Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome.

1 2( , , , )rn n nn satisfies 1 2 .rn n n n

1

n

n

1

2

n n

n

1 2

3

n n n

n

1 2 1r

r

n n n n

n

1

1np 2

2np 3

3np rn

rp

31 21 1 21 2 3

1 1 2 1 2 3 1 2 3

( )! ( )!!

!( )! !( )! !( )!rnn n n

r

n n n n nnp p p p

n n n n n n n n n n n n


?( )p X n


probabilities p1, p2, …, pr such that1

1r

iip

Define X=(X1, X2, …, Xr) st Xi is the number of trials that resulted in the ith outcome.

1 2( , , , )rn n nn satisfies 1 2 .rn n n n

1

n

n

1

2

n n

n

1 2

3

n n n

n

1 2 1r

r

n n n n

n

1

1np 2

2np 3

3np rn

rp

1 21 2

1 2

!

! ! !rn n n

rr

np p p

n n n

31 2

1 2 31 2

rnn n nr

r

np p p p

n n n

Example 26

If a pair of dice are tossed 6 times, what is the probability of obtaining a total of 7 or 11 twice, a matching pair one, and any other combination 3 times?

If a pair of dice are tossed 6 times, what is the probability of obtaining a total of 7 or 11 twice, a matching pair one, and any other combination 3 times?

Three outcomes:

1. 7 or 11

2. match

3. others

1 ?p

2 ?p

3 ?p

2 / 9

1/ 6

11/18

X1 #7 or 11;

X2 #matches;

X3 #others.

2 3

1 2 3

6 2 1 11( 2, 1, 3)

2 1 3 9 6 18P X X X

2 36! 2 1 11

2!1!3! 9 6 18

0.1127

Sums of Independent Variables

Generating Functions


The Sum of Independent Random Variables

X Y 1 2

1 2

( ) { , , }

( ) { , , }

I X x x

I Y y y

Z X Y ) ?(Zp z

( ) ( )Zp z P Z z ( )P X Y z

,i iiP X x Y z x ( , )i i

i

P X x Y z x ( ) ( )i i

i

P X x P Y z x ( ) ( )X i Y i

i

p x p z x

Example 27

Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).


?( )I Z X Y {0,1,2, , 2 }n

[ ]X Y z [0 ][0 ][ ]x n y n x y z

[0 ][0 ][ ]x n y n y z x

[0[0 ] [ ]]zx n z xx yn

[[0 [ ]]] z x n zx n y z x

[0 ][ ][ ]x n z n x z y z x [ ]X Y z

2

,

1( , ) ( ) ( )

1X Y X Yp x y p x p yn

, 0,1, ,x y n

Example 27



[0 ][ ][ ]x n z n x z y z x [ ]X Y z

0 n

z n z

0 n

z n z

Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n}

[ ] [0 ][ ]X Y z x z y z x [ ] [ ][ ]X Y z z n x n y z x

?( )I Z X Y {0,1,2, , 2 }n

, 2

1( , ) ( ) ( )

( 1)X Y X Yp x y p x p yn

, 0,1, ,x y n

Example 27



Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n}


?( )I Z X Y {0,1,2, , 2 }n

, 2

1( , ) ( ) ( )


, 0,1, ,x y n

,0

( ) ( , )z

X Yx

P X Y z p x z x

,( ) ( , )n

X Yx z n

P X Y z p x z x

2

1

( 1)

z

n

2

2 1

( 1)

n z

n

Example 27



Case 1: z{0, 1, …, n} Case 2: z{n+1, n+2, …, 2n}


?( )I Z X Y {0,1,2, , 2 }n

, 2

1( , ) ( ) ( )


, 0,1, ,x y n

,0

( ) ( , )z

X Yx

P X Y z p x z x

,( ) ( , )n

X Yx z n

P X Y z p x z x

2

1

( 1)

z

n

2

2 1

( 1)

n z

n

2

2

10,1, ,

( 1)( )

2 11, 2, , 2

( 1)

zz n

nP X Y z

n zz n n n

n

2

2

10,1, ,

( 1)( )

2 11, 2, , 2

( 1)

zz n

nP X Y z

n zz n n n

n

Probability Generating Functions

Probabilities

Probabilities

機率母函數


Let X be a nonnegative integer-valued random variable. Its probability generating function GX(t) is defined as:

0 1 2( ) (0) (1) (2) ( )X X X Xx

Xt t t tG t p p p p x

0

( ) ( ) xX X

x

G t p x t

pgf



0 1 2( ) (0) (1) (2) ( )X X X Xx


0

( ) ( ) xX X

x

G t p x t

pgf

0011

22

xx

?( )P X x



0 1 2( ) (0) (1) (2) ( )X X X Xx


0

( ) ( ) xX X

x

G t p x t

pgf

0011

22

xx

?( )P X x

Probability Generating Functions0

( ) ( ) xX X

x

G t p x t

pgf

Compute the pgf’s for the following distributions:1. X ~ B(n, p); 2. Y ~ P(); 3. Z ~ G(p); 4. U ~ NB(r, p).



0

( ) ( ) xX X

x

G t p x t

pgf



( ) (1 ) , 0,1, ,x n xX

np x p p x n

x

0

( ) (1 )n

x xn xX

x

nG t p p t

x

0

( ) (1 )n

x n x

x

np pt

x

( 1 )npt p

( )npt q 1q p


0

( ) ( ) xX X

x

G t p x t

pgf



( ) , 0,1,2,!

y

Y

ep y y

y

0

( )!

y

Yy

yte

G ty

0

( )

!

y

y

et

y

te e

( 1)te


0

( ) ( ) xX X

x

G t p x t

pgf



1( ) (1 ) , 1, 2,zZp z p p z

1

1

( ) (1 )z zZ

z

tG t p p

1

0

(1 ) zz

z

tp p

0

(1 )z z

z

p pt t

0

(1 )z

z

p pt t

1 (1 )

p

p

t

t

1

p

q

t

t

1q p


0

( ) ( ) xX X

x

G t p x t

pgf



( )1

r

U

pG

t

tt

q

1q p

Exercise

Important Generating Functions

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

Theorem 2 Sums of Independent Random Variables

Let X, Y be two independent, nonnegative integer-valued random variables. Then,

( ) ( ) ( )X Y X YG t G t G t


( ) ( ) ( )X Y X YG t G t G t

X Y ( ), ( ) {0,1,2, }I X I Y and

Let Z=X+Y.Pf)( ) ( )X Y ZG t G t

0

( ) zZ

z

tp z

00, ( , )

z

X Yx

z

z

tp x z x

0 x z

0 z

0 x z

0 x

x z 0

( ) ( ) z

x z xX Yp z x tx p

0

( ) ( )x z x

x z xX Yt tp x p z x

0 0

( ) ( )Xx z

x zY z tp x t p

( ) ( )X YG t G t


( ) ( ) ( )X Y X YG t G t G t

X Y ( ), ( ) {0,1,2, }I X I Y and

Fact:

1 2 1 2( ) ( ) ( ) ( )

n nX X X X X XG t G t G t G t

1 2( ), ( ), , ( ) {0,1,2, }nI X I X I X and1 2 nX X X. . .

Example 29

Example 27 Let X, Y be two independent random variables each uniformly distributed over 0, 1, 2, …, n. Find P(X+Y = z).


Use pgf to recompute Example 27.Use pgf to recompute Example 27.

( ) ( )X YG t G t11 1

1 1

nt

n t

0 11

1nt t t

n

( ) ( ) ( )X Y X YG t G t G t 1 2 22

1(1 ) (1 )

( 1)nt t

n

1 2 22

0

11(1 2 )

( 1)n n z

z

zt t t

zn

1 2 22

0

1(1 2 ) ( 1)

( 1)n n z

z

t t z tn

Example 29



Use pgf to recompute Example 27.Use pgf to recompute Example 27.

( ) ( )X YG t G t11 1

1 1

nt

n t

0 11

1nt t t

n

( ) ( ) ( )X Y X YG t G t G t 1 2 22

1(1 ) (1 )

( 1)nt t

n

1 2 22

0

11(1 2 )

( 1)n n z

z

zt t t

zn

1 2 22

0

1(1 2 ) ( 1)

( 1)n n z

z

t t z tn

2

2

10,1, ,

( 1)( )

2 11, 2, , 2

( 1)

zz n

nP X Y z

n zz n n n

n

2

2

10,1, ,

( 1)( )

2 11, 2, , 2

( 1)

zz n

nP X Y z

n zz n n n

n

Theorem 3

Theorem 3

1 2 rX X X

表何意義 ?~ ?( , )B r p

Theorem 31 2 1 2

( ) ( ) ( ) ( )r rX X X X X XG t G t G t G t

1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .

1 2 rX X X. . .

1 2 rX X X ~ ?

) ?(iXG t ( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

1 2( ) ?

rX X XG t

pt q

rpt q

( , )B r p

Theorem 3

1 2 rX X X

表何意義 ?

1 2~ ( ?, )rB n n n p

1 2 1 2( ) ( ) ( ) ( )

r rX X X X X XG t G t G t G t 1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .

1 2 rX X X. . .

Theorem 3

1 2 rX X X ~ ?) ?(

iXG t

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

1 2( ) ?

rX X XG t

inpt q

1 2 rn n npt q

1 2 1 2( ) ( ) ( ) ( )

r rX X X X X XG t G t G t G t 1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .1 2( ), ( ), , ( ) {0,1,2, }rI X I X I X and1 2 rX X X. . .

1 2 rX X X. . .

1 2( , )rB n n n p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X

表何意義 ?, )?~ (NB r p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X ~ ?

) ?(iXG t

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

1 2

( ) ?rX X XG t

1

pt

qt

1

rpt

qt

( , )NB r p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X

表何意義 ?1 2~ ( , )?rNB p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X ~ ?) ?(

iXG t

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

1 2( ) ?

rX X XG t

1

ipt

qt

1 2

1

rpt

qt

1 2( , )rNB p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X

表何意義 ?1 2~ ( ?, )rP p

Theorem 31 2 1 2



1 2 rX X X. . .

1 2 rX X X ~ ?

) ?(iXG t

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

( 1)

~ ( , ) ( ) ( )

~ ( ) ( )

~ ( ) ( )1

~ ( , ) ( )1

nX

tX

X

r

X

X B n p G t pt q

X P G t e

ptX G p G t

qt

ptX NB r p G t

qt

1 2( ) ?

rX X XG t

( 1)i te

1 2( )( 1)r te

1 2( , )rP p

Theorem 3熟記 !!! 請靈活的將它們用於解題

Functions of Multiple Random

Variables


Functions of Multiple Random Variables

Let X, Y be two random variables with jpmf pX,Y(x, y).

Suppose that 1

2

( , )

( , )

U

V

g X Y

g X Y

1-1 pU,V(u, v)=?



Suppose that 1-1 pU,V(u, v)=?

Example:

U

V

X Y

X Y

X $/month Y $/month

pX,Y(x, y) 已知

pU,V(u, v) = ?

1

2

( , )

( , )

U

V

g X Y

g X Y




Example:

U

V

X Y

X Y

pX,Y(x, y) 已知

pU,V(u, v) = ?

1-1 implies invertible.

1

2

( , )

( , )

X h

Y h

U V

U V

2

2

U V

U V

X

Y

1

2

( , )

( , )

U

V

g X Y

g X Y




1-1 implies invertible.

1

2

( , )

( , )

X h

Y h

U V

U V

, ( , ) ( , )U Vp Pu v U u V v

1 2( , ), ( , )P X h Y vhu v u

, 1 2( , ), ( , )X Y u vp h uh v

1

2

( , )

( , )

U

V

g X Y

g X Y

Example 30

Let X~B(n, p1), Y~B(m, p2) be two independent random variables.

U = X + YV = X Y


U = X + YV = X Y

Let Find pU,V(u, v).

Example 30


U = X + YV = X Y


U = X + YV = X Y

Let Find pU,V(u, v).

, 1 2 1 2( , ) (1 ) (1 )x y n x m yX Y

n mp x y p p p p

x y

X Y

2

2

U V

U V

X

Y

, , 2 2( , ) ,X Yu v u v

U V u vp p

2 2 2 21 2 1 2

2 2

(1 ) (1 )u v u v u v u vn

u v u v

mn mp p p p

0,1, ,

0,1, ,

x n

y m

, , 1 2( , ) ( , ), ( , )X YU Vp p h v hu v u u v , , 1 2( , ) ( , ), ( , )X YU Vp p h v hu v u u v

{0,1, , }u n m { , , }v m n

2 {0,1, , }u v n

2 {0,1, , }u v m and

Documents

Chapter 3-2 Discrete Random Variables 主講人 : 虞台文. Content Functions of a Single Discrete Random Variable Discrete Random Vectors Independent of Random