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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.
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"Square root" of Beta(a,b) distribution
I might have misunderstood the question but, if not, I suggest this.
Denoting, $Z = X \ Y$, take Mellin-Transform expectations of both-sides.
Since $E[X^{s-1}] = B(a+s,b)$, this condition you demand …