For $0<s_1<s_2<t$ evaluate conditional expected value $$E[N\left( s_1 \right) N\left( s_2 \right)|N\left(t\right)],$$ where $N\left( t\right)$ is Poisson process.
Here is what I've got. By using the independence of increments of the process and CEV properties, I modified it to $$E[N(s_1)|N(t)]E[N(s_2)-N(s_1)|N(t)]+E[N^2(s_1)|N(t)]$$ Then, I evaluated the first multiplier in the first term as $$E[N(s_1)|N(t)=n]=\sum^n_{k=0}kP(N(s_1)=k|N(t)=n).$$ That led me to $$E[N(s_1)|N(t)=n]=E[Bin(n, \frac{s_1}{t})]=n\frac{s_1}{t}.$$ The same I evaluated the second multiplier. The problem is I can not handle the second term, i.e. $E[N^2(s_1)|N(t)]$. What do I need to do with it?
$N(s_1)|N(t)=k$ is a Binomial random variable with $n=k$ and $p = \frac{s_1}{t}$. Now since $Var(X) = E(X^2)-E(X)^2$, $E(X^2) = Var(X)+E(X)^2$ and since the variance of a Bernoulli random variable is $np(1-p)$, it follows:
$E(N^2(s_1)|N(t)=k) = k(\frac{s_1}{t})(1-\frac{s_1}{t}) + k^2\frac{s^2_1}{t^2}$