Page 70 - DMTH404_STATISTICS

Statistics

Notes Hence the density of Z is given by 4
1

z (z ) =  f(z ,z )dz
1 1 1 2 2


1 
f (z  z )f (z )dz ,    z ,z ,   ...(3)
2
1
2
2
1
2
2

This formula giving the density function of Z1 is known as the convolution formula. This is
called the convolution formula because the density function is the convolution product of the
density functions of X and X .
1 2
Let us now dlculate the distribution function of Z . We denote the distribution function of Z by
1 1
z . Then we have
1
z
z (z) =  1  (z )dz
1 1 1

z   
1 
=   f (z  z )f (z )dz 2  dz 1
1
2
2
2
  
   
=   f (z  z )dz 1 f (z )dz 2
1 
1
2
2
2
  
  z 2 z 
=    f (u)du f (z )dz 2

2
2
1
     
(by the transformation u = z - z )
1 2

1 

= F (z z )f (z )dz 2
2
2
2

where F is the distribution function of X .
1 1
Therefore the distribution function of Z is the convolution product of the distribution function
1
of X and the density function of X .
1 2
The above relation gives an explicit formula for the distribution function of Z .
1
Let us see an ,example.
Example 5: Suppose X and X are independent random variables with the gamma
1 2
distributions having parameters ( , ) and ( , ) respectively. Let us find the density function
l 2
of the sum Z = X + X using the convolution formula.
1 2
The density of X is
1
 i  x  1 e  l i x
Fx (x ) = i , x > 0
i i i
(   i )
= 0 otherwise
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