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If $X$ and $Y$ are two Beta random variables, I am interested in the distribution of their ratio $X/Y$. More specifically, I am interested in the moment generating function of this ratio. There is a paper of Pham-Gia that apparently computes the distribution but I don't have access to it and I don't know how helpful it will be for determining the moment generating function. What is known about these?

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  • $\begingroup$ I've located the paper, so now the only question is: what is known about the moments of this ratio? $\endgroup$ – Aryeh Kontorovich Nov 7 '13 at 10:05
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OP wrote:

what is known about the moments of this ratio?

I have not seen the paper ... but one does not even need to derive the distribution of the ratio in order to derive the moments of the ratio. In particular:

If $X$ ~ $Beta(a,b)$ and $Y$ ~ $Beta(c,d)$ are independent, then the joint pdf of $(X,Y)$ is, say, $f(x,y)$:

(source)

Then, the $k$-th raw moment of the ratio $\frac{X}{Y}$ can be derived immediately as:

(source)

where I am using the Expect function from the mathStatica add-on to Mathematica to automate the nitty-gritties for me (I am one of the developers of the former). If desired, one can express the solution slightly more neatly as:

$$\frac{B(a+k,b) B(c-k,d)}{B(a,b) B(c,d)}$$

where $B$ denote the Euler beta function.

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