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The Exact Density Function of a Sum of Independent Gamma Random Variables as an Inverse Mellin Transform

Iman Mabrouk, Serge B. Provost

Abstract


Sums of gamma random variables have been utilized for modeling purposes in connection with problems arising in engineering, queueing theory and communication theory, among other areas of application. Particular attention is paid in this paper to linear combinations of chi-square random variables, as quadratic forms in central normal vectors possess such a structure and several test statistics encountered in multivariate analysis are asymptotically so distributed. The inverse Mellin transform technique is employed to derive the exact density function of a weighted sum of independently distributed chi-square random variables. Three numerical examples attest to the validity and applicability of the resulting representation of the density function.

Keywords


sums of gamma random variables, quadratic forms, linear combinations, chi-square random variables, generalized gamma distribution, moments, generalized hyperge-ometric functions, exact density function, inverse Mellin transform technique.

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