Consider some variable $X\sim \operatorname{Poi}(\lambda(t))$ to be Poisson-distributed with some parameter $\lambda$ dependent on time, where we know how the random variable $\lambda$ is distributed.
How is the expected value and the variance of $X$ computed for a given time window of length $T$? That is, considering the measurements of a random variable $X$ over some time window $T$, what is the expected value of the group of observations and its variance?
Just answering this for others searching (even though the current question has relatively low views).
The only magic in computing an inhomogeneous Poisson process in a given time comes from computing the second moment. This is done quite easily by noting that, for some random parameter $\lambda_t$, the second moment if given by (for $N_T\sim \text{Poi}(\lambda_t, T)$ for an interval $T$)
$$ \begin{align} \left\langle N_T^2\right\rangle &= \sum_{n\ge 0}\sum_{\vartheta\in S_\lambda}n^2\frac{e^{-\vartheta}\vartheta^n}{n!}P(\lambda = \vartheta) \\ &= \sum_{\vartheta\in S_\lambda}\vartheta e^{-\vartheta}P(\lambda = \vartheta)\sum_{n\ge 0}\frac{n\vartheta^{n-1}}{(n-1)!} \\ &= \sum_{\vartheta\in S_\lambda}\vartheta e^{-\vartheta}P(\lambda = \vartheta)\left(\sum_{n\ge 0}\frac{(n-1)\vartheta^{n-1}}{(n-1)!}+\sum_{n\ge 0}\frac{\vartheta^{n-1}}{(n-1)!}\right) \\ &= \sum_{\vartheta\in S_\lambda}\vartheta e^{-\vartheta}P(\lambda = \vartheta)\left(\vartheta e^{\vartheta} + e^{\vartheta}\right)\\ &= \left\langle \lambda_t^2\right\rangle + \left\langle \lambda_t\right\rangle \end{align} $$
Where $S_\lambda$ is the support of $\lambda_t$. Additionally, I dropped some dependencies on the time interval in question, but that shouldn't be a terrible problem as this can be filled in quite easily.