intuition beyond a Beta property

betas

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?

One Response to “intuition beyond a Beta property”

  1. 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.

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