Poisson-processes and it's arrival times I am currently studying for my non-life insurance exam and have the following problem:
Let $S(t) = \sum_{i=1}^{N(t)} (X_i + T_i)^2$, where $X_i$ are i.i.d. r.v. with density $f(x)$ and $T_i$ are the arrival times of the homogeneous possion process $N(t)$ with intensity $\lambda =2$. With a fiven density $f(x) = \exp(-x)$ for $x \geq 0$, how can one calculate $E[S(t)]$?
Now I know that $P(T_1 > t) = \exp(-\int_0^t \lambda(s) ds) = \exp(-2t)$. So the density would be given by $g_1(t) = 2\exp(-2t) $.
Furthermore I could write the following:
$$ S(t) = \sum_{i=1}^{N(t)} (X_i + T_i)^2 = \sum_{i=1}^{N(t)} X_i^2 + 2\sum_{i=1}^{N(t)} X_i T_i + \sum_{i=1}^{N(t)}T_i^2 $$
If I would have only $\sum_{i=1}^{N(t)} X_i^2$ I'd know that
$$ E[S(t)] = E[S(t) \mid N(t)] = E[N(t)]E[X_i^2] $$
How can I proceed with the arrival times?

http://www.maths.qmul.ac.uk/~ig/MAS338/PP%20and%20uniform%20d-n.pdf
Using the well-known result about the symmetric functional of the arrival times
(Theorem 1.2), we have
$$ \begin{align} E[S(t)] &= E\left[\sum_{i=1}^{N(t)} (X_i + T_i)^2\right] \\
&= E\left[\left[\sum_{i=1}^{N(t)} (X_i + T_i)^2\Bigg|N(t)\right]\right] \\
&= E\left[\left[\sum_{i=1}^{N(t)} (X_i + U_i)^2\Bigg|N(t)\right]\right] \\
&= E[N(t)]E[(X_1+U_1)^2] \\
\end{align}$$
where $U_i$ are i.i.d. as $\text{Uniform}(0, t)$ and independent of $N(t)$. Now we just remain to calculate
$$ E[X_1] = 1, E[X_1^2] = 2, E[U_1] = \frac {t} {2}, E[U_1^2] = \frac {t^2} {3}$$
and thus
$$ E[N(t)]E[(X_1+U_1)^2] 
= \lambda t\left(2 + 2 \times 1 \times \frac {t} {2} + \frac {t^2} {3}\right)
= \frac {\lambda t} {3}(t^2 + 3t + 6) $$
