Variance of Inhomogenous Poisson process in a given window 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?
 A: 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.
