Chapter 4

Joint Distribution &

Function of rV

Joint Discrete Distribution

Definition

Xxxxx

xXxXxXPxxxf

XXX

k

k

kkk

k

of ),...,,( valuespossible all

,...,,,...,,

be todefined is ),...,,(X variablerandom discrete

ldimensiona- theof pdf)(joint function density y probabilitjoint The

21

221121

21

1),...,,(.2

,...,, valuespossible allfor 0,...,,.1

: satisfied are properties following theifonly if ),...,,(X rV

valued- vectorsomefor pdfjoint theis ,...,,function A

1 2

21

2121

21

21

x x

k

kk

k

k

xxxf

xxxxxxf

XXX

xxxf

Joint discrete CDF

kkk

k

xXxXxXPxxxf

XXX

k

,...,,,...,,

be todefined is ),...,,(X variablerandom discrete

ldimensiona- theof cdfjoint The

221121

21

Example for joint distributions

Consider the following table:

Using the table, we have

Y=0 Y=3 Y=4

X=5 1/7 1/7 1/7 3/7

X=8 3/7 0 1/7 4/7

4/7 1/7 2/7

pX

pY

.7/37/27/1)4(p)3(p 3YP

7/3)5(p7XP

2/7p(5,4)p(5,3)3Y7,XP

YY

X

The Marginal PDF

1

2

2122

2111

21

2121

,)(

,)(

are and of spdf' marginal then the

),...,,( pdfjoint thehas rV discrete of pair theIf

x

x

k

xxfxf

xxfxf

XX

xxxf,XX

Example : Air Conditioner Maintenance– A company that services air conditioner

units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner

– The random variable X, taking the values 1,2,3 and 4, is the service time in hours

– The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units

Expected Values for Jointly Distributed Random VariablesExpected Values for Jointly Distributed Random Variables

Let X and Y be discrete random variables with joint probability density function p(x, y). Let the sets of values of X and Y be A and B, resp. We define E(X) and E(Y) as

For the random variables X and Y from the previous slide example,

).(pE(Y) and )(pE(X)B

YA

X yyxxyx

.7

47

7

48

7

35E(X)

.7

11

7

24

7

13E(Y)

• Joint p.d.f

• Joint cdf

Y=number of units

X=service time

1 2 3 4

1 0.12 0.08 0.07 0.05

2 0.08 0.15 0.21 0.13

3 0.01 0.01 0.02 0.07

1

07.0...08.012.0

i j

ijp

43.0

08.015.008.012.02,2

F

Find E[X] and E[Y] !!

Previously Example– Marginal p.d.f of X

– Marginal p.d.f of Y

3

11

( 1) 0.12 0.08 0.01 0.21jj

P X p

4

11

( 1) 0.12 0.08 0.07 0.05 0.32ii

P Y p

Joint continuous distribution

),...,,( allfor

,...,,,...,,

: as written becan

CDFjoint that theasXsuch of pdfjoint thecalled

,...,,function a is thereif continuous be tosaid is

),...,,(X rV valued- vectorldimensiona-A

21

212121

21

21

k

k

x x

kk

k

k

xxxx

dtdtdttttfxxxF

xxxf

XXXk

k k

kA k

k

dxdxdxxxxfAXP

XXXk

2121

21

,...,,

: have A weevent ldimensiona-k a and

),...,,(X rV valued- vectorldimensiona-A

Th

1),...,,(.2

,...,, valuespossible allfor 0,...,,.1

: satisfied are properties following theifonly if ),...,,(X rV

valued- vectorsomefor pdfjoint theis ,...,,function A

21

2121

21

21

k

kk

k

k

xxxf

xxxxxxf

XXX

xxxf

Ex

10,10,4),(by given is pdfjoint that theAssume 212121 xxxxxxf

a. Then find the joint CDF !

5.0

2 Find b. 21 XXP

The Marginal Continuous PDF

12122

22111

21

2121

,)(

,)(

are and of spdf' marginal then the

),...,,( pdfjoint thehas rV scontinuuou of pair theIf

dxxxfxf

dxxxfxf

XX

xxxf,XX k

Find )( and )( 2211 xfxf From previously example

Ex

)(xf

xxxcxxxf

XXX

33

321321

321

findthen

10,,,

form theof pdfjoint a with continuous be ,,Let

Independent rV

k

iiikkk

ii

k

bXaPbXabXaf

ba

XXX

11111

21

,...,

every if

t independen be tosaid are ,,, Variables Randomt Independen Def.

)()(,,

)()(,,

: holds properties following ifonly if

t independen be tosaid are ,,, Variables Random Th.

111

111

21

kkk

kkk

k

xfxfxxf

xFxFxxF

XXX

Conditional pdf

Joint MGF

0h somefor and ,..., where

)(

: be todefined is existsit if of MGFjoint The

1

1

1

hthttt

eEtM

,...,XXX

ik

Xt

X

k

k

iii

Group

Discuss the exercise bellow !

BAIN, page : 166-

No 7, 9, 21, 30

Time : 30’