Given a Poisson arrival process of rate $\lambda$, and an observation of $k$ arrivals during an (arbitrary externally imposed) window of duration $t$, and prior p.d.f. $p(\lambda)$, is the following (unnormalized) posterior p.d.f. correct?
\begin{align}p(\lambda\mid t,k) &= \frac{p(k\mid t,\lambda)p(t,\lambda)}{p(k,t)}\\ \\ &= \frac{p(k\mid t,\lambda)p(t\mid \lambda)p(\lambda)}{p(k,t)}\\ \\ &\propto_\lambda \frac{(\lambda t)^k}{k!}e^{-\lambda t}p(\lambda) \\ &\propto_\lambda \frac{t^k}{k!}\lambda^ke^{-\lambda t}p(\lambda) \\ &\propto_\lambda \lambda^ke^{-\lambda t}p(\lambda). \end{align}