Let $X_i$ be a sequence of iid random variables, with de density function $f_X(x)=\mathrm{e}^{-x}$. Further, we have a uniform Poisson process $N(t)$ with an intensity $\lambda=2$, and hitting times $T_i$. The question is to calculate the expectation of $\sum_{i=1}^{N(t)} (X_i+T_i)^2$.
I know that for the uniform poisson process, the expectation of such a sum can be easily calculated if the summands are iid, which is here not the case, since $T_i$ is Gamma distributed with $\alpha=i$ and $\beta=\lambda$. Howewer, I also know that $T_i=W_1+...+W_i$, where $W$ is iid and exponentially distributed with mean $\lambda=2$.
I would appreciate any hints and ideas how to approach this problem.