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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
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