Solution
= Solution
The likelihood is $\theta e^{-\theta X}$. Multiplication by the prior gives a density proportional to $\theta^m e^{-(\lambda+X)\theta}$, so
$$
\theta\mid X\sim\Gamma(m+1,\lambda+X)
$$
in shape-rate parametrisation.
= Solution
The likelihood is $\theta e^{-\theta X}$. Multiplication by the prior gives a density proportional to $\theta^m e^{-(\lambda+X)\theta}$, so
$$
\theta\mid X\sim\Gamma(m+1,\lambda+X)
$$
in shape-rate parametrisation.