Fnding the distribution of functions of random variables

There are three techniques for finding the distribution of functions of
random variables will be presented. These three techniques are called (1) the cumulative-distribution-function technique,
(2) the moment-generating-function technique,
(3) the transformation technique,
important examples are given, including the distribution of sums of independent random variables
For instance, the distributions of chi•square, Student s t, and F, all derived from sampling from a normal distribution,
necessary to solve the problem of finding the distribution of the function of the given random variables. One important function of given random variables is their sum.
Distribution of Sums of Independent Random Variables
We employ the moment-generating-function technique to find the distribution of the sum of independent random variables.
We found the exact distribution of the sums of certain independent random variables. Other examples, including the important result that the sum of independent identically distributed geometric random
variables has a negative binomial distribution, are given in the Problems.
Note, however, that if the distribution of the sum is known, then the distribution of the average is readily derivable.