2 Probability Theory 2.3 Generating and Sampling Two Signed Bernoulli Random Variables with Given Correlation 2.5 Distribution of the Determinant of a Complex-Valued Sample Correlation Matrix

2.4 Distribution of A Simple Function of Gamma Variables

In this post, we look at a nifty result presented in [11] where the probability density function (pdf) of the function $\dfrac{r_{1}}{c+r_{2}}$ two independent Gamma distributed random variables $r_{1}\sim\mathcal{G}(k_{1},\theta_{1})$ and $r_{2}\sim\mathcal{G}(k_{2},\theta_{2})$ is derived.

The derivation is an exercise (for the scalar case) in computing the pdf of functions of variables by constructing the joint pdf and marginalizing based on the Jacobian. A similar approach can also be used for matrix-variate distributions (which will probably be a good topic for another post.)

Theorem 6.

Let $c>0,$ and let $r_{1}\sim\mathcal{G}(k_{1},\theta_{1})$ and $r_{2}\sim\mathcal{G}(k_{2},\theta_{2})$ be independent random variables, then, the pdf of $r=\dfrac{r_{1}}{c+r_{2}}$ , denoted by $p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c),$ is given by

K_{\mathrm{r}}r^{k_{1}-1}\exp\left(-\frac{rc}{\theta_{1}}\right)U\left(k_{2},k% _{1}+k_{2}+1;c\left(\frac{r}{\theta_{1}}+\frac{1}{\theta_{2}}\right)\right),

(2.42)

where

U(a,b;z)=\frac{1}{\Gamma(a)}\int_{0}^{+\infty}\exp(-zx)x^{a-1}(1+x)^{b-a-1}% \mathrm{d}x

is the hypergeometric U-function [15, Chapter 13, Kummer function], and $K_{\mathrm{r}}$ is a constant ensuring that the integral over the pdf equals one.

Proof.

As $r_{1}$ and $r_{2}$ are independent, their joint pdf, denoted by $p_{r_{1},r_{2}}(r_{1},r_{2};k_{1},\theta_{1},k_{2},\theta_{2}),$ is given by

\frac{1}{\Gamma(k_{1})\theta_{1}^{k_{1}}\Gamma(k_{2})\theta_{2}^{k_{2}}}r_{1}^% {k_{1}-1}r_{2}^{k_{2}-1}\exp\left(-\frac{r_{1}}{\theta_{1}}-\frac{r_{2}}{% \theta_{2}}\right).

(2.43)

Applying transformation $r=\frac{r_{1}}{c+r_{2}}$ with Jacobian $\frac{\mathrm{d}r_{1}}{\mathrm{d}r}=c+r_{2},$ we obtain the transformed pdf, $p_{r,r_{2}}(r,r_{2};k_{1},\theta_{1},k_{2},\theta_{2},c)$ , as

\displaystyle K_{\mathrm{r}}^{\prime}r^{k_{1}-1}(c+r_{2})^{k_{1}}r_{2}^{k_{2}-% 1}\exp\left(-\frac{rc}{\theta_{1}}-\left(\frac{r}{\theta_{1}}+\frac{1}{\theta_% {2}}\right)r_{2}\right),

(2.44)

where $K_{\mathrm{r}}^{\prime}$ is a constant ensuring that the integral over the pdf equals one. Next, $p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c)$ is obtained by marginalization as

p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c)=\int_{0}^{+\infty}p_{r,r_{2}}(r,r% _{2};k_{1},\theta_{1},k_{2},\theta_{2},c)\mathrm{d}r_{2},

(2.45)

where the integration is conducted using the integral representation of the hypergeometric U-function from [15, Chapter 13, Kummer function] to obtain the expression in the theorem statement. ∎

Version History

1.

First published: 19th Oct. 2021 on aravindhk-math.blogspot.com
2.

Modified: 17th Dec. 2023 – Style updates for LaTeX