Let X|θ have a Gamma distribution Gamma(α, β). Assume that the shape parameter α is

known.

(a) Show that Gamma distribution belongs to the Exponential Family of Distributions with

respect to the parameter β.

(b) Find the conjugate prior of the Gamma distribution.

(c) Find the conjugate posterior density of β.

The Theorem we are given I have attached

I have solved 2a)

and for 2b) I haveβ^(αB) x e^(βA)

but is apparently incorrect

If someone could go through this, that would be brilliant it is my first post so let me know if you need any more information