Let $$S(t)=\sum_{i=1}^{N(t)}(X_i+T_i)^2$$ where $N(t)$ is a homogenous Poisson process with rate $\lambda=2$, $X_i$ iid with density $f(x)=e^{-x},x\geq0$ and $T_i$ are the arrival times of $N(t)$. $T_i$ and $X_i$ are independent.
I want to determine $\mathbb E[S(t)]$.
We have $\mathbb E[X_i]=1$, $\mathbb E[X_i^2]=2$.
How do can I evaluate $\mathbb E[T_i]$ and $\mathbb E[T_i^2]$?
The final result for $\mathbb E[S(t)]$ shoud be $\mathbb E[S(t)]=\mathbb E[N(t)]\Big(\mathbb E[X_i^2]+2\mathbb E[X_i]\mathbb E[T_i]+\mathbb E[T_i^2]\Big)=2(2+2\mathbb E[T_i]+\mathbb E[T_i^2]\Big)=??$