Abstract
A sum of gamma random variables is a mathematically tractable approach to approximate multiple sources when the power (amplitude squared) of a single source is nearly gamma distributed because the sum can be expressed analytically for a wide variety of cases. Previous work indicates that the gamma distribution is a good two-parameter empirical approximation of received power from a single, elevated sound source in a turbulent atmosphere. The gamma distribution can be conceptualized as a sum of k independent and identically distributed exponential random variables, each with mean m. Here, k and m are called the shape and scale parameters, respectively. Thus, the summation of multiple gamma random variables with the same scale parameter still yields a gamma distribution. When multiple sources can be approximated well using a gamma distribution, then those sources could be grouped together for simplicity and conceived as a single source in a turbulent atmosphere. In addition, multiple authors have derived analytic expressions for the sum of N independent gamma random variables with distinct parameters in terms of the confluent hypergeometric functions. If the gamma random variables are correlated, then the result is still analytic using recurrence relations.
Published Version
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