Let $N(t)$ have Poisson distribution with parameter $\lambda t $ (meaning, expected value of $\lambda$ events per hour. What is the expectation of:
$N(6) \cdot N(10)$? (The expected value of the number of events in 6 hours multiplied by the number of events in 10 hours)
What I've tried is:
$E(N(6) \cdot N(10))=E(N(6)\cdot (N(6)+N(7,10)))=E(N(6)^2 + N(7,10)))=E(N(6)^2)+E(N(6)E(N(7,10)))=E(N(6)^2)+6\lambda \cdot 3\lambda$
But I got stuck calculating $E(N(6)^2)$
$$ {\rm E}[N(6)^2]=\mathrm{Var}(N(6))+{\rm E}[N(6)]^2 $$