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Exact Distribution of the Ratio of Gamma and Rayleigh Random Variables

The distributions of the ratio of two independent random variables arise in many applied problems and have been extensively studied by many researchers. This article derives the distributions of the ratio Z=|x/y| , when x and y are gamma and Rayleigh random variables respectively and are distributed independently of each other. The associated pdf, cdf, and moments have been given in terms of different special functions, for examples, confluent hypergeometric function, parabolic-cylinder function and beta functions. Some plots for the cdf and pdf associated with the distribution of the ratio have been provided.