## intuition beyond a Beta property

**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?

*Related*

This entry was posted on March 30, 2015 at 12:15 am and is filed under Books, Kids, R, Statistics, University life with tags beta distribution, cross validated, moment generating function, Stack Echange. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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March 30, 2015 at 12:50 am

Maybe try √x = cos(θ), y = sin2(φ), or something like that?

Michael Betancourt put a little note on the arXiv a while ago that used the trick of taking the square root of jointly Dirichlet-distributed random variables. The square root transformation maps the simplex to the part of the surface of a sphere that’s in the all-positive orthant, and it’s very natural to then transform to spherical co-ordinates.