# intuition beyond a Beta property

**Xi'an's Og » R**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**A** self-study question on X validated exposed an interesting property of the Beta distribution:

If x is B(n,m) and y is B(n+½,m) then √xy is B(2n,2m)

While this can presumably be established by a mere change of variables, I could not carry the derivation till the end and used instead the moment generating function E[(XY)^{s/2}] since it naturally leads to ratios of B(a,b) functions and to nice cancellations thanks to the ½ in some Gamma functions [and this was the solution proposed on X validated]. However, I wonder at a more fundamental derivation of the property that would stem from a statistical reasoning… Trying with the ratio of Gamma random variables did not work. And the connection with order statistics does not apply because of the ½. Any idea?

Filed under: Books, Kids, R, Statistics, University life Tagged: beta distribution, cross validated, moment generating function, Stack Echange

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