I am having issues trying to solve this exercise in Bayesian analysis.
The waiting time in minutes until being serviced by a phone call center follows an Exponential(λ) model, with E[y|λ] = 1/λ. Out of a sample of n independent waiting times, one is only provided with information about the largest waiting time in the sample, y ≥ 0, which has a probability density function equal to: $$p(y|λ) = nλ(1 − e^{-λy})^{n-1}e^{−λy}$$ where the parameter space can be assumed to be equal to Ω = [0, ∞). Experts agree on choosing as a prior distribution for λ a Gamma(a = 10, b = 10) distribution, with pdf: $$b^a λ^{a−1}e^{−bλ}\overΓ(a)$$ π(λ) = Γ(a) , (2) and hence with E[λ] = $a\over b$ and V ar[λ] = $a\over b^2$. One is told that the largest waiting time in a sample of n = 10 independent waiting times has been 2.42 minutes.
What is the statistical model for the largest waiting time in the sample, what is the Bayesian model and what is the likelihood function?
How would you compute the posterior distribution for λ?
So far I did the first point but I am not sure about it. I used as a bayesian model $$p(y|λ) = nλ(1 − e^{-λy})^{n-1}e^{−λy}$$ using the Gamma as a priori. For the likelihood I take this expression with y=2.42. Is it correct? I can't menage to get the posterior distribution because the integral I get seems to me too difficult to be solved analitically.
Please help me with some hints, thank you