Let and be independent random variables having
the respective probability density functions and .
Then the cumulative distribution function of the random
variable can be given as follows.

where is the cdf of .
By differentiating , we can obtain the pdf of as
The form of integration
is called the *convolution*.
Thus, the pdf is given by the convolution of
the pdf's and .
However, the use of moment generating function
makes it easier to ``find the distribution of the sum of independent
random variables.''
Let and be independent normal random variables
with the respective parameters
and
.
Then the sum of random variables has the mgf

which is the mgf of normal distribution with parameter
.
By the property (a) of mgf,
we can find that
is a normal random variable
with parameter
.
Let and be independent gamma random variables
with the respective parameters
and
.
Then the sum of random variables has the mgf

which is the mgf of gamma distribution with parameter
.
Thus, is a gamma random variable
with parameter
.

Generated by MATH GO: 2006-02-27