Suppose that $X_1, ... , X_n$ each have an exponential distribution with parameter $\theta$, and suppose that the prior for $\theta$ is an exponential distribution with parameter $\lambda$. Find the posterior distribution of $\theta$.

I have to solve the above problem but I'm getting stuck. I've calculated the likelihood function $f(x|\theta)$ to be $\theta^nexp(-\theta r)$ where $r$ is the sum of the $x_is$. Then the prior has pdf $\pi (\theta) = \lambda exp(-\lambda\theta)$.

Then I'm getting a posterior which is proportional to $\lambda \theta^n exp(-\theta (\lambda + r))$, but I don't see where to go from here. Usually the posterior looks like a distribution I recognise but here, I'm not sure how to find the normalisation constant.

Any help would be greatly appreciated. Thanks


The function you calculated can be written as $$λθ^ne^{-(λ+r)θ}=λθ^{(n+1)-1}e^{-(λ+r)θ}$$ which is (if I am not mistaken) a gamma distribution with shape parameter $n+1$ and scale parameter $θ=(λ+r)^{-1}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.