In class, our professor told us to verify this solution on our own time. The problem is:
Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of $X(t)$, given that $X(t+s) = n$.
and the answer he gave us is:
$$\binom{n}{k}\dfrac{t^k s^{n-k}}{(t+s)^n}$$
I'm having a hard time understanding how he came up with that answer, any suggestions?
The basic points are that $A = X(t)$ and $B = X(s+t)-X(t)$ are independent Poisson random variables with parameters $\lambda t$ and $\lambda s$ respectively, and $A+B$ is a Poisson random variable with parameter $\lambda (t+s)$. For $0 \le k \le n$, $$P(A=k, B = n-k) = P(A=k) P(B=n-k) = e^{-\lambda t} \dfrac{(\lambda t)^k}{k!} e^{-\lambda s} \dfrac{(\lambda s)^{n-k}}{(n-k)!}$$ $$P(A+B=n) = e^{-\lambda(t+s)} \dfrac{(\lambda (t+s))^n}{n!}$$ Divide and simplify...