BIVARIATE DISTRIBUTION

CHAPTER 3

WEEK 5

2

3.5 Marginal Probability Functions

Definition:

1. If X and Y are discrete random variables with joint

probability function f(x,y), then the marginal probability

functions of X and Y are

( ) ( ) ( , )X

y

f x P X x f x y

( ) ( ) ( , )Y

x

f y P Y y f x y

3

2. If the joint probability density function of continuous

random variables X and Y is f(x,y), the marginal probability

density functions of X and Y are

where Rx denotes the set of all points in the range of (X,Y)

for which X = xwhere x is fixed and y varies and Ry

denotes the set of all points in the range of (X,Y) for which

Y = ywhere y is fixed and x varies.

( ) ( , ) x

X

R

f x f x y dy

( ) ( , ) y

Y

R

f y f x y dx

4

Example 3.11:

Using the joint probability distribution (Ex.3.9)as follows:

1. Find the marginal probabilit y function of X,

2. Find the marginal probability function of Y ,

f(x,y) 0 1 2

0 1/18 1/9 1/6

1 1/9 1/18 1/9

2 1/6 1/6 1/18

X=x

Y=y

xfX

yfY

5

Solution:

a) ( ) ( ) ( , )X

y

f x p X x f x y 2

0

(0) (0, ) (0,0) (0,1) (0,2)X

y

f f y f f f

3

1

6

1

9

1

18

1

2

0

(1) (1, ) (1,0) (1,1) (1,2)X

y

f f y f f f

3

1

6

1

18

1

9

1

2

0

(2) (2, ) (2,0) (2,1) (2,2)X

y

f f y f f f

3

1

18

1

9

1

6

1

6

b) ( ) ( ) ( , )Y

x

f y p Y y f x y

2

0

(0) ( ,0) (0,0) (1,0) (2,0)Y

x

f f x f f f

3

1

6

1

9

1

18

1

2

0

(1) ( ,1) (0,1) (1,1) (2,1)Y

x

f f x f f f

18

5

9

1

18

1

9

1

2

0

(2) ( , 2) (0,2) (1,2) (2,2)Y

x

f f x f f f

18

7

18

1

6

1

6

1

7

Example 3.12:

Using the joint probability density function from Example 3.10,

find

a) the marginal probability function of X, fX(x).

b) the marginal probability function of Y, fY(y).

otherwise ;0

1y0 ,10 ;4,

xxyyxf

8

Solution:

a)

b)

( ) ( , )X

Rx

f x f x y dy

xxydy

y

y

24

1

0

10 x

( ) ( , )Y

Ry

f y f x y dx

1

0

24

x

x

yxydx 10 y

9

Example 3.13:

The joint probability density function for random variables Y1

and Y2 is given as follows:

a) Find the marginal probability density function of

b) Find the marginal probability density function of

1 2( )

1 2

1 2

; 0 , 0( , )

0 ; elsewhere

y ye y yf y y

11 1, ( )YY f y

22 2, ( ).YY f y

10

Solution:

a)

1 1 1 2 2

0

( ) ( , )Yf y f y y dy

2

2

21

0

2

)(

y

y

yydye

)( 21 yyu 12

dy

du2dydu

21 2

12

( )1 0 0

0

( )yy yu u

Y yf y e du e e

10y

e

1ye

10 y

11

b)

2 2 1 2 1

0

( ) ( , )Yf y f y y dy

0

1

)( 21 dyeyy

2ye

20 y

12

Example 3.14:

The joint probability density function of X and Y is given as

follows:

a) Find p.

b) Find the marginal probability density function of X, fX(x).

d) Find the marginal probability density function of Y, fY(y).

2 2( ) ; 0 2, 1 4 ( , )

0 ; elsewhere

p x y x yf x y

13

b)

, 0<x<2

c) , 1<y<4

4

2 2

1

1( ) ( )

50Xf x x y dy

50

213 2

x

2 22 2

0

8 61( ) ( )

50 150Y

yf y x y dx

4

1 0

22 1)(2 dxdyyxP

1

50p

14

Quiz 3.5:

If X is the amount of money(in Ringgit) that a housewife spends

on petrol during a day and Y is the corresponding amount of

money(in Ringgit) for which she reimbursed from her husband and

the joint probability of these two random variables is given by

and

Find the marginal density of X.

1 20( , ) for 10 20,

25 2

x xf x y x y x

x

( , ) 0; elsewheref x y

15

3.6 Independence

Definition:

Let X and Y denote the random variables of either continuous

or discrete type which have the joint probability distribution

function f(x,y) and marginal probability density functions of

fX(x) and fY(y) respectively. The random variables X and Y are

said to be independent if and only if

f(x,y) = fX(x) fY(y)

16

Example 3.15:

The joint probability distribution of X and Y is given in a table

below:

f(x,y) 1 2 3 4 5 6

1 1/36 1/36 1/36 1/36 1/36 1/36

2 1/36 1/36 1/36 1/36 1/36 1/36

3 1/36 1/36 1/36 1/36 1/36 1/36

4 1/36 1/36 1/36 1/36 1/36 1/36

5 1/36 1/36 1/36 1/36 1/36 1/36

6 1/36 1/36 1/36 1/36 1/36 1/36

17

a) Find the marginal probability function for X.

b) Find the marginal probability function for Y.

c) Are X and Y independent? Prove it.

18

Solution:

a)

b)

c)

For all x,y ,

.: X and Y are independent.

x 1 2 3 4 5 6

g(x) 1/6 1/6 1/6 1/6 1/6 1/6

Y 1 2 3 4 5 6

h(y) 1/6 1/6 1/6 1/6 1/6 1/6

361)1,1( f

361)1()1( hg

)().(),( yhxgyxf

19

Example 3.16:

The joint probability distribution of X and Y is given as below:

a) Find the marginal probability function for X.

b) Find the marginal probability function for Y.

c) Are X and Y independent? Prove it.

f(x,y) 1 2 31 1/9 1/9 1/92 1/9 1/9 1/93 1/9 1/9 1/9

20

Solution:

a)

b)

c)

.: X

and Y are independent.

x 1 2 3

g(x) 1/3 1/3 1/3

Y 1 2 3

g(y) 1/3 1/3 1/3

91)1,1( f

9

1

3

1.

3

1)1().1( hg

)1().1()1,1( hgf

)().(),( xhxgyxf

21

Example 3.17:

The joint probability density function is given as follows:

a)Find the marginal probability density function for X.

b)Find the marginal probability density function for Y.

c) Are X and Y independent? Prove it.

2 ; 0 1, 0 1( , )

0 ; elsewhere

x x yf x y

22

Solution:

a) ;

0 < x < 1

b) ;

0 < y < 1

c)

.: X and Y are independent.

1

1

00

( ) 2 2 2y

X

y

f x xdy xy x

1

12

00

( ) 2 1x

Y

x

f y xdx x

xyxf 2),( ( ). ( ) 2X Yf x f y x

( , ) ( ). ( )X Yf x y f x f y

23

Example 3.18:

The joint probability density function is given as follows

Are X and Y independent? Prove it.

; 0 1, 0 1( , )

0 ; elsewhere

x y x yf x y

24

Solution:

.: X and Y are not independent.

11 2

0 0

1( ) ( ) ; 0 1

2 2X

yf x x y dy xy x x

11 2

0 0

1( ) 0 ( ) ; 0 1

2 2Y

xf y x y dx xy y y

1 1

( , ) ( )( ) ( ). ( )2 2

X Yf x y x y x y f x f y

25

Quiz 3.6:

Given X and Y are two random variables that are

independent. Show that

( , ) ( ) ( )F x y F x F y