## Sum of independent random variables

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