# what will be the distribution of ratio of correlated gamma distributed random variables?

If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes standard gamma probability density function and '$\sim$' represents 'distributed as'. X and Y are independent random variables.

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This looks like it's going to get messy. How far have you gotten in your own attempts to find the pdf of $R$? – Mike Spivey Oct 22 '10 at 15:54
I assumed $S=X+Y$ and find the joint pdf $f_{R,S}(r,s)$. Later, I averaged over S taking limit '0' to 'infinity' and got $f_R(r)$, but not able to find the valid range of R to verify this pdf. – dikuve Oct 22 '10 at 16:44
It seems like the range of values that $R$ can take would be $(0, \infty)$. The range of values of $\frac{X}{X+Y}$ is $(0,1)$, independent of $X$ and $Y$, and since $X$ and $Y$ can be as small or as large as you please $\frac{C}{X+Y}$ would have a range of $(0, \infty)$. – Mike Spivey Oct 22 '10 at 17:10
The title says "correlated gamma distributed random variables" while the last sentence of the text says that $X$ and $Y$ are independent. Which is correct? Also, are $a$ and $\sigma_x^2$ the mean and the variance of $X$? Gamma random variables are often specified in terms of the order parameter $t$ and the scale parameter $\lambda$ and $\Gamma(t, \lambda)$ or $\Gamma(t, 1/\lambda)$ is used to denote a Gamma distributed random variable with mean $t/\lambda$. For one case of the ratio of correlated Gamma random variables, see here – Dilip Sarwate Nov 2 '11 at 11:17
Yes its correct as title says "Ratio of correlated gamma...." because it is ratio of X+C to X+Y where X random variable is related to both numerator and denominator whether X and Y are independent r.v. – dikuve Dec 16 '11 at 17:14