I'm trying to teach myself Bayesian statistics and am currently trying find the posterior distribution on the following problem:
Suppose $X_1,...,X_n$ are iid exponential random variables, with density $f(x;\theta) =\theta e^{-\theta x}$ ,and let us suppose that we have a prior on $\theta$ with density $\theta^{\alpha -1}e^{-\beta \theta}$
Now, to find the posterior, we do Likelihood times Prior. The likelihood is given by
$$\theta^n e^{-n^2\theta \bar{x}}$$
So that we have
$$\theta^{\alpha -1}e^{-\beta \theta}\theta^n e^{-n^2\theta \bar{x}}=\theta^{n+\alpha-1}e^{-n^2 \theta \bar{x} - \beta \theta}$$
And in the textbook I am reading, from here you are meant to deduce the posterior distribution, but I cannot see an obvious distribution. Have I made an error? Can the posterior density function contain the random variable (x in this case)?