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Thread: Linear Exponential Family & Inverse Exponeital Distribution

1. Linear Exponential Family & Inverse Exponeital Distribution

Suppose the model distribution X|Λ=λ is inverse exponential
f(x|λ) = λ exp(-λ/x) / x^2
and the prior distirbution for Λ is gamma with parameters α and θ.
(note: this belongs to the case of "exact credibility", so that the Bayesian and Buhlmann premiums are the same.)
In particular, the model distribution is in the linear exponential family, i.e. f(x|λ) = p(x) exp[r(λ) x] / q(λ).
Determine p(x), r(λ), and q(λ).

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Can anyone please explain how to solve this problem? It seems like I'm missing something because I have no idea how to get the exp[r(λ) x] part in this problem. How can I make the x to appear at the top when it is actually at the bottom for the inverse exponential density?

Any help would be greatly appreciated!