Can anyone help me with this Q about algebra on gamma distributions?

If the pdff(x)ofXis given by a gamma distributiongamma(shape, scale)

...and the pdfg(y)ofYis given by(A/X^2) * f(A/Y)

...where Y is X*N

...what I want is a parametric form forg(y). Ithinkit 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 thatf(A/Y)is an inverse gamma distributiongammainv(shape, newscale)

...wherenewscale = A/(N*scale)

...(note this isf(A/Y),notf(A/X)which would be a gamma distributiongamma(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))

wherenewscale2 = (A / (Y/N)^2 ) * newscale

= (A / (Y/N)^2) * ( A / (N*scale) )

=A^2 / ( (Y/N) * scale )

this could be completely wrong of course...

Ta!