Consider a time Interval $[0,T]$ and times $0<t_1 < t_2 < ... < t_n<T$ generated by a Poisson process.
In my scriptum, the expected value of the function $$Y(t) = \sum_{t_i \leq t} \kappa (t-t_i)$$ is computed, where $\kappa$ is some arbitraty kernel, e.g. $\kappa(t-t_i)=\exp(-(t-t_i))$.
The scriptum states:
If zero events occur, we have $E[Y]=0$, if one event occurs, we have $$E[Y]=p_\lambda(1) \int_0^T p_1(t) \kappa(T-t) dt$$ with $$p_\lambda(N)= \frac{e^{-\lambda T} (\lambda T)^N}{N!} $$
I have great difficulty understanding this! Here are my problems:
1) The only things that makes $Y$ a random variable is the times $t_i$. But if we consider them fixed before we start our calculation, $Y$ is not a rv anymore. Hence, we can't consider them fixed, but $t_i$ themselves are random variables. Correct?
2) Wouldn't it be mathematically correct to write $$Y(t,\omega)=\sum_{t_i(\omega) \leq t} \kappa(t-t_i(\omega)),$$ where $t_i(\omega)$ are random variables? But how are they distributed then?
3) Can one give me any hint how to mathematically calculate the expectation of $Y$ and hence justify the calculations above?
The text is indeed faulty. One way out is to introduce the (random) number of events up to time $T$, say $N_T$, and to consider the (elementary) conditional expectations $E(Y\mid N_T=n)$ for various values of $n$. For example, $E(Y\mid N_T=0)=0$ and $$E(Y\mid N_T=1)=K,\qquad K=\int_0^t\kappa(t-s)\mathrm ds=\int_0^t\kappa(s)\mathrm ds,$$ hence $$E(Y\,;\,N_T=1)=P(N_T=1)K=p_\lambda(1)K.$$ (Not sure, though, that I understand the appearance of $p_1(t)$ in the integral in your question.)
Likewise, conditionally on $N_T=n$ with $n\geqslant1$, the collection of points $\{t_1,\ldots,t_n\}$ is distributed as $n$ i.i.d. points uniformly distributed in $[0,T]$ hence $$E(Y\mid N_T=n)=nK,$$ and finally, $$E(Y)=\sum_{n=0}^\infty nKp_\lambda(n)=K\mathrm e^{-\lambda T}\sum_{n=0}^\infty n\frac{(\lambda T)^n}{n!}=(\lambda T)K.$$