Following Kevin Murphy's MLAPP and I am having troubles with Exercise 3.11. The exercise happens in parts, and I am hoping to get help with multiple parts.
A lifetime $X$ of a machine is modeled by an exponential distribution with unknown parameter $\theta$. The likelihood is $$ p(x \mid \theta) = \theta e^{-\theta x} $$
Assume an expert believes $\theta$ should have a prior distribution that is also exponential $$ p(\theta) = \text{Expon}(\theta \mid \lambda) = \text{Gamma}(\theta \mid 1, \lambda) = \lambda e^{-\lambda \theta} $$
Note: $\mathcal{D} = \{ x_1, \dots, x_N \}$ is the data.
1. What is the posterior $p(\theta \mid \mathcal{D}, \lambda)$?
I get $$ \begin{align} p(\theta \mid \mathcal{D}, \lambda) &= p(\theta \mid \lambda) p(\mathcal{D} \mid \theta) \\ &= \lambda e^{-\lambda \theta} \prod_{i=1}^N \theta e^{-\theta x_i} \\ &= \lambda \theta^N e^{-\theta (\lambda + \sum_{i=1}^N x_i)} \end{align} $$
which doesn't look like any distribution that I know of.
2. Is the exponential prior conjugate to the exponential likelihood?
No. The posterior is not an exponential. But I'm not sure if I computed the posterior correctly.
I assume that I'm doing something wrong, because I'm later asked about the posterior mean, which is difficult to do by integration (integration by parts and extracting a term that depends upon the Gamma function).