On linear combinations of independent exponential variables

The exact distribution of a linear combination of n indepedent negative exponential random variables , when the coefficients cf the linear combination are distinct and positive , is well-known. Recently Ali and Obaidullah (1982) extended this result by taking the coeff icients to be arbitrary real numbers. They used a lengthy geometric.

al approach to arrive at the result . This article gives a simple derivation of the result with the help of a generalized partial fraction technique. This technique also works when the variables involved are gamma variables with certain types of parameters. Results are presented in a form which can easily be programmed for computational purposes. Connection of this problem t o various problems in different fields is also pointed out.