A geometric interpretation of the relations between the exponential and generalized Erlang distributions

Linear combinations of exponential distribution functions are considered, and the class of distribution functions so obtainable is investigated. Convex combinations correspond to hyperexponential distributions, while non-convex combinations yield, among other, generalized Erlang distributions obtainable as sums of independent exponential random variables with different parameters. For a given number n of different exponential distributions, the class investigated is an ( n – 1)-dimensional convex subset of the n -dimensional real vector space generated by the n distribution functions. The geometric aspect of this subset is revealed, together with the location of hyperexponential and generalized Erlang distributions.