Suppose the waiting time in a queue is modeled as an exponential random variable with unknown parameter , and that the average time to serve a random sample of 20 customers is 5.1 minutes. A gamma distribution with mean 0.5 and variance 1 is used as the prior. Find the posterior distribution and mean.

So I worked out the prior distribution to be , and got to the point where

Problem is (and this seems really silly), I don't even know what the "waiting time" given the data is supposed to be. I guessed it might be something like 2.4225, or 2.55 depending on how you calculate it, and after that use a substitution to evaluate the bottom integral I guess?