Suppose the waiting time in a queue is modeled as an exponential random variable with unknown parameter \lambda, 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 Gamma~(0.25,0.5), and got to the point where

f(\lambda| x) = \frac{\lambda^{0.25}{e^{-\lambda(0.5+x)}}}{\int^{\infty}_0 \lambda^{0.25}{e^{-\lambda(0.5+x)}}d\lambda}

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?