Can anyone help me with this Q about algebra on gamma distributions?
If the pdf f(x) of X is given by a gamma distribution gamma(shape, scale)
...and the pdf g(y) of Y is given by (A/X^2) * f(A/Y)
...where Y is X*N
...what I want is a parametric form for g(y). I think it should be an inverse gamma distribution, and I just need to do some algebra to find the parameters.
...now first Q: am I correct in thinking that f(A/Y) is an inverse gamma distribution gammainv(shape, newscale)
...where newscale = A/(N*scale)
...(note this is f(A/Y), not f(A/X) which would be a gamma distribution gamma(shape, A/scale)
...which comes from two properties of gamma distributions, according to wikipedia:
- if X = gamma(scale, shape), 1/X = invgamma(shape, 1/scale)
- if X = invgamma(scale,shape), cX = invgamma(shape, c*scale)
...second Q: how should I now add the pre-multiplication by (A/X^2) to this?
Multiply the scale parameter again, directly?
....(A/X^2) * f(A/Y) = g(y) = gammainv(shape, newscale2))
where newscale2 = (A / (Y/N)^2 ) * newscale
= (A / (Y/N)^2) * ( A / (N*scale) )
= A^2 / ( (Y/N) * scale )
this could be completely wrong of course...