Find Find $ E \{ \frac{ \xi(5) }{ \xi(7)+1 } \ | \xi(7) \} $ where $ \xi(t) $ is standard Poisson process
I can easily solve similar problems, where a random process is represented by an ordinary function, for example Xcos (t + Y), where X \in N(0, 1) and Y \in R(-\pi, \pi) considering mathematical expectation of a section. But in this case, I can’t realize what I need to start solution. Of course, I know all theory material about Poisson process and Wiener process too, and I have similar task for it: Find $ E \{ W(1) | W(3) \} $ where $ W(t) $ is standard Wiener process.
I don’t need a solution, just give me a hint for start it. Thanks in advance.
By denoting $x = \xi(7)$ you get the denominator is simply $x+1$, so your only task is to find $\Bbb E[\xi(5)|\xi(7) = x]$. By definition you have that $$ \Bbb E[\xi(5)|\xi(7) = x] = \sum_{k=0}^\infty k\cdot \Bbb P(\xi(5) = k|\xi(7) = x) $$ so you need to compute the latter conditional probabilities. I hope you are good with $\Bbb P(\xi(7) = x|\Bbb \xi(5) = k)$ and you know Bayes' rule.