# Algebra on gamma distributions

• Oct 19th 2012, 03:44 AM
JohnGriffiths
Algebra on gamma distributions
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:

1. if X = gamma(scale, shape), 1/X = invgamma(shape, 1/scale)
2. 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...

Ta!
• Oct 19th 2012, 05:35 PM
chiro
Re: Algebra on gamma distributions
Hey JohnGriffiths.

You have written a lot down and its a bit hard to decipher: Can you just point out in the simplest way what distribution you have (call it X) and what new random variable you want (i.e f(X)) by stating the transformation of the variable itself without adding all the extra stuff.
• Oct 19th 2012, 07:18 PM
MaxJasper
Re: Algebra on gamma distributions
Check if the following CDF for g(y) is of any use for x>=0:

$\displaystyle \text{CDFg}(X\geq x)\text{:=}\frac{\int_0^X \frac{n \theta ^{-k} \left(\frac{a}{n x}\right)^k e^{-\frac{a}{\theta n x}}}{x \Gamma (k)} \, dx}{n \theta ^{-k} \left(\frac{a}{n}\right)^k \left(\frac{a}{\theta n}\right)^{-k}}$(Shake)
• Oct 22nd 2012, 02:03 PM
JohnGriffiths
Re: Algebra on gamma distributions