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Let $X_1,\ldots, X_n$ are iid $\mathrm{Gamma}(\alpha,\beta)$, $Y_1,\ldots, Y_n$ are iid $\mathrm{Gamma}(\alpha,\gamma)$ and independent of $X_i$. What will be the distribution of $\frac{\bar X}{\bar Y}$?

I know the ratio of two independent gamma random variates is a beta prime distribution (when rate parameters are fixed and only shape parameters varying of two gamma variates) {source}.

Also sum of two independent gamma random variates, when rate parameters are fixed and only shape parameters varying of two gamma variates, is another gamma variate whose shape parameter is the sum of shape parameters of the two gamma variates and common rate parameter (source).

But in my case, shape parameters are fixed and rate parameters varying and also it is ratio of two sample mean.

How can I derive the distribution of $\frac{\bar X}{\bar Y}$?

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ABC
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    $\bar X/\bar Y = \frac{(1/n)(X_1+\dotsb+X_n)}{(1/n)(Y_1+\dotsb+Y_n)} = X/Y$, where $X$ is the sum in the numerator and follows $\text{Gamma}(n\alpha,\beta)$, and similar for $Y\sim \text{Gamma}(n\alpha, \gamma)$. Then use your usual technique to find the distribution of the quotient of two random variables. – Em. May 27 '16 at 00:52
  • Letting $U_1=\frac{X}{Y}$ and $U_2=Y$, I find $$f_{U_1,U_2}(u_1,u_2)=\frac{{\beta}^{n\alpha}}{\Gamma(n\alpha)}\frac{{\gamma}^{n\alpha}}{\Gamma(n\alpha)}u_1^{n\alpha-1}u_2^{n\alpha-1}exp(-u_1u_2\beta)exp(-u_2\gamma).$$ How can I integrate with respect to $u_1$ and $u_2$? – ABC May 27 '16 at 01:59
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    Ratio distribution. – Em. May 27 '16 at 02:02

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