The Exact Distribution of Sums Weights of Gamma Variables

We consider a representation of the probability density function of a weighted convolution of the gamma distribution, where a confluent hypergeometric function describes how the differences between the parameters of the components of scale lead to departures from a density range. It is shown that the distributions can be characterized as the product between a gamma density and a confluent hypergeometric function. We give closed-form expressions for the cumulative, survival and hazard rate function. The corresponding moment generating function(m.g.f) and cumulant generating function(c.g.f) have been calculated and their properties have bean discussed