Page 71 - DMTH404_STATISTICS
P. 71
Unit 5: Functions of Random Variables
for i = 1, 2. We use Formula (3) to mmpute the density function of 2. For z > 0, we have Notes
z(z) = fx (z u)fx (u)du
1
2
= fx (z u)fx (u)du 2
1
0
z 1 e (z u) 2 e u
= (z u) 1 1 u 2 1 du.
0 ( 1 ) ( 2 )
1 2 z z
e
= {(z u) 1 1 u 2 1 du}
( 1 ) ( 2 ) 0
1 2 2 1 1
= e z 1 (1 v) 1 1 v 2 1 dv
z
( 1 ) ( 2 ) 0
u
(by the transformation v = )
z
1 2 z 1 2 1
= e z B( 2 , 1 )
( ) ( )
1 2
1 2
z
z
\ z(z) = e z 1 2 1 , 0
( )
1 2
= 0 , z < 0.
The last equality follows from the properties of beta function and gamma function.
This example shows that the convolution of gamma distributions with parameters ( , ) and
l
( , ) is a gamma distribution with parameter ( + , ).
2 1 2
Next we shall consider another example in which we illustrate another method called Moment
Generating Function approach. This method is useful for finding the distribution functions of
sums or linear combinations of independent random variables.
Example 6: Suppose X and X are independent random variables with distributions
1 2
2
2
N[ , ] and N[ , ] respectively. Define Z = X + X . Then the m.g.f. of Z is
1 1 2 2 1 2
Mz (t) = E [e t(X 1 + X 2 ) ]
= E [e e ]
tX l
tX 2
= E[e ] E[e ]
tX 1
tX 2
The last relation follows from the fact that e and e lX 2 are independent random variables when
tX l
XI and X2 are independent. But we have proved earlier that
LOVELY PROFESSIONAL UNIVERSITY 63
