Exact Distributions of the Linear Combination of Gamma and Rayleigh Random Variables

The distribution of a linear combination of random variables arise in many applied problems, and have been extensively studied by different researchers. This article derived the exact distribution of the linear combination aX + bY , where a > 0 and b are real constants, and X and Y denote gamma and Rayleigh random variables respectively and are distributed independently of each other. The associated cdfs and pdfs have been derived. The plots for the cdf and pdf, percentile points for selected coefficients and parameters, and the statistical application of the results have been provided. We hope thefindings of the paper will be useful for practitioners in various fields.