Distribution and Moments of a Normalized Dissimilarity Ratio for two Correlated Gamma Variables

Abstract

We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $\rho$. Our focus is on the parameter: \[ D(X,Y) = \frac{|X - Y|}{X + Y}, \] which appears in applied contexts such as dynamic speckle imaging, where it is known as the \textit{Fujii index}. In this work, we derive a closed-form expression for the probability density function of $D(X,Y)$ as well as analytical formulas for its moments of order $k$. Our derivation starts by representing $X$ and $Y$ as two correlated exponential random variables, obtained from the squared magnitudes of circular complex Gaussian variables. By considering the sum of $k$ independent exponential variables, we then derive the joint density of $(X,Y)$ when $X$ and $Y$ are two correlated Gamma variables. Through appropriate varable transformations, we obtain the theoretical distribution of $D(X,Y)$ and evaluate its moments analytically. These theoretical findings are validated through numerical simulations, with particular attention to two specific cases: zero correlation and unit shape parameter.