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Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$.

I found

$$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + \Gamma(b)} \left(\frac{w}{w+1}\right)^a \left(\frac{1}{w+1}\right)^b \frac{1}{w} $$ So,

$$ f_W(w) = \operatorname{dbeta}\left(\frac{w}{w+1}, a+1, b+1 \right) \frac{1}{w} $$

or

$$ f_{X/Y}(x/y) = \operatorname{dbeta}\left(\frac{x}{x+y}, a+1, b+1 \right) \frac{y}{x} $$

Is there any story or interpretations behind this result? I know that

$$ \frac{X}{X+Y} \sim \operatorname{Beta}(a,b), $$

but how does this relate to $X/Y$?

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2 Answers 2

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Based on the comments, I figured out that its the so-called beta prime distribution, or beta distribution of the second kind (http://en.wikipedia.org/wiki/Beta_prime_distribution).

The connection to the beta distribution is: If $X \sim \operatorname{Beta}(\alpha, \beta)$, then $\frac{X}{1-X} \sim \operatorname{Beta'}(\alpha, \beta)$.

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Notice that $X+Y$ is the sum of two gamma distributed random variables, thus $(X+Y) \sim Γ(a+b, λ)$. You can see this from the moment generating function of $X+Y$.

So, $\dfrac{X}{Y}$ and $\dfrac{X}{X+Y}$ are not really that different; they are both ratios of gamma distributed random variables.

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