ch3ch3 MultionariateMultionariate

Multionariate Random VariablesMultionariate Random Variables

Random VariablesRandom Variables

3.23.2 ProbabilityProbabilitymass/density functions

Probability mass/density functions

,....2,1,),( jiPyYxXP ijii

0)1( ijP 1)2(1 1,

i jij

jiij pP

a) Discrete Multivariate r.v

Definition 3.4

Suppose the r.v. X and Y are discrete.

is defined by

The joint pmf

XY 21 ixxx

jy

y

y

2

1 12111 ippp

22212 ippp

21 ijjj ppp

,....2,1,),( jiPyYxXP ijii

The Joint pmf of X and Y

selectedballsblackofnumberX selectedballswhiteofnumberY

Example 3.1

A bag contains 3 black, 2 white and 1 red balls. 2 balls

Find the joint probability mass function YandXof

are chosen at random without replacement. Let

Solution:

yYxXPyxF ,),(

yy xx

ij

j i

p

The Joint cdf of X and Y

Example 3.2

XY 21

2

1 310

3131

The joint pmf of X and Y is given by

Find the cdf of X and Y

Solution:

1j

ijyall

jii p)yY,xX(P)xX(Pj

The marginal pmf of X and Y

1i

ijxall

jij p)yY,xX(P)yY(Pi

,....2,1,)()(11

jipyYPandpxXPi

ijjYj

ijiX

Definition 3.5

Suppose the r.v. X and Y are discrete.

pmf is defined by

The marginal

;,2,1i,p}xX{P)yY,xX(P)x(P1j

ijially

jiiX

j

.,2,1j,p}yY{P)yY,xX(P)y(P1i

ijjallx

jijY

i

XY ixxx 21

jy

y

y

2

1 12111 ippp

22212 ippp

ijjj ppp 21

1

, ),(),()(j xx

ijYXX

i

pxFYxXPxF

The marginal cdf of X and Y

1

, ),(),()(i yy

ijYXY

i

pyFyYXPyF

XY ixxx 21

jy

y

y

2

1 12111 ippp

22212 ippp

ijjj ppp 21

selectedballsblackofnumberX selectedballswhiteofnumberY

Example 3.3 (Example 3.1)

A bag contains 3 black, 2 white and 1 red balls. 2 balls

Find the joint pmf and marginal pmf of YandX

are chosen at random without replacement. Let

Solution:

b) Continuous Multivariate r.v.b) Continuous Multivariate r.v.

1) Double integrals1) Double integrals

R

dxdyyxfvolume ),(

How do we calculate the double integral above?

R

b

a

xg

xgdxdyyxfdxdyyxf ]),([),(

)(

)(

2

1

)()( 21 xgYandxgY bxa ① R is a region bounded by the curves

where

)()( 21 yhxandyhx ② R is a region bounded by the curves

dyc where

R

yx dxdyeV

Ryxeyxf yx ),(),(

10,2 xandyxy

Example 3.4

Let R is the triangle bounded byDefine

Find the volume

2),(, RBandyx

B

YX dxdyyxfBYXP ),()),(( ,

2) Joint probability density functions

Definition 3.6),(, yxf YXThe nonnegative function is a joint pdf of

the continuous random variable X and Y if

for all

The properties of f (x , y):2),(0),()1( , Ryxyxf YX

1),(, dxdyyxf YX

,(( , ) ) ( , )X YB

P X Y B f x y dxdy

(2) The relationship between the joint cdf and pdf is

ordudvvufyxFx y

YXYX ),(),( ,,

),()],([ ,,

2

yxfyxFyx YXYX

),(, yxF YX),(, yxf YXi.e. the joint pdf is the derivative of

with respect to x and y.

(3) We can calculate the probability that (X,Y) falls ina rectangle as below

,(( , ) ) ( , )X YB

P X Y B f x y dxdy

(3) We can calculate the probability that (X,Y) falls ina rectangle as below

b

a

xg

xgdxdyyxfxgyxgbxaP ]),([))()(,(

)(

)(21

2

1

b

a

d

c YX dxdyyxfdycbxaP ]),([),( ,

d

c

yh

yhdydxyxfdycyhxyhP ]),([)),()((

)(

)(21

2

1

More generally More generally

The marginal cdf of X and Y

dxdyyxfxFxFx

YXX

)),((),()( ,

dyyxfxf YXX ),()( ,

dxyxfyf YXY ),()( ,

The marginal pdf of X and Y

dxxfx

X )(

.d),()( ,

yyxfxf YXX

dydxyxfyFyFy

YXY

)),((),()( , dyyf

y

Y )(

.d),()( ,

xyxfyf YXY

Let X and Y denote the proportion of time out of one

Example 3.5

working day that two employees, performing their assigned tasks.

Y is given by

)4/1,2/1()1( YXPFind

w

yxyxyxf YX .00

10,10),(,

)1()2( YXPFind

(3) Find the marginal pdf of X and Y.

A and B, spendThe joint pdf of X and

.d),()(

yyxfxfX

联合分布 边缘分布

.d),()(

xyxfyfY

小 结小 结

.dd),(),()(

y

Y yxyxfyFxF

.dd),(),()(

x

X xyyxfxFxF

Solution: The joint pmf of X and Y is given by

0 1 20 1 2

00 11 22

0 3/15 3/150 3/15 3/15

2/15 6/15 02/15 6/15 0

1/15 0 01/15 0 0

xy

selectedballsblackofnumberX selectedballswhiteofnumberY

A bag contains 3 black, 2 white and 1 red balls. 2 balls

Find the joint probability mass function YandXof

are chosen at random without replacement. Let

XY 21

2

1 310

3131

Solution:

21

1

2

o x

y

)2,2()2,1(

)1,1( )1,2(

,1yor1xWhen)1( ),( yxF },{ yYxXP

;0

),( yxF

,2y1,2x1When)2(

11p ;0

21

1

2

o x

y

)2,2()2,1(

)1,1( )1,2(

XY 21

2

1 310

3131

,2y,2x1When)3(

),( yxF 1211 pp ;31

,2y1,2xWhen)4(

;31),( 2111 ppyxF

,2y,2xWhen)5( ),( yxF

22122111 pppp

.1

So,the cdf of ( X ,Y ) is given by

),( yxF ;0

),( yxF ;0

),( yxF ;31

),( yxF .1

(1) 1 1 ,When x or y

(2) 1 2,1 2,When x y

(3) 1 2, 2,When x y

(5) 2, 2,When x y

(4) 2,1 2,When x y ( , ) 1 3;F x y

0, 1 1,

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

31, 2, 2.

x or y

F x y x y or x y

x y

0 1 20 1 2

00 11 22

0 3/15 3/150 3/15 3/15

2/15 6/15 02/15 6/15 0

1/15 0 01/15 0 0

xy

The joint pmf of X and Y is given bySolution:

selectedballsblackofnumberX selectedballswhiteofnumberY

A bag contains 3 black, 2 white and 1 red balls. 2 balls

Find the joint pmf and marginal pmf of YandX

are chosen at random without replacement. Let

0 1 20 1 2

00 11 22

0 3/15 3/150 3/15 3/15

2/15 6/15 02/15 6/15 0

1/15 0 01/15 0 0

xy

The marginal distributions of X and Y arex

)( xXP 15

3

15

9210

15

3

y

)( yYP 15

6

15

8210

15

1

0 1 20 1 2

00

11

22

0 0 3/153/153/15 3/15

006/156/152/152/15

00001/151/15 15

1

15

8

15

6

15

3

15

9

15

31)xX(P i

)yY(P j

The joint pmf and marginal pmf of X and Y is given by

Joint Marginal

+

+ +

+

+

+

+ +

+ +

+ +

j,i

ijp

xy