Poisson Process

Question

Suppose $\{N(t), t\geq 0\}$ is a Poisson proces with rate $\lambda$. For $s<t$, find: $\mathbb{E}[N(s)\mid N(t)=\alpha]$. The answer is $ \frac{\alpha s}{t}$. I intuitively get the answer, but I do not know how to get to it mathematically, could someone help?

Answer 1

The short way (use your words)

Because the Poisson events are independent, and occur at a constant average rate, when given that $\alpha$ have happened in a certain period of length $t$, then the count of events in any period of length $s$ that is a subset of that interval, will have a binomial distribution.   $N(s)\sim \mathcal {Bin}(N(t), s/t)$.   Hence the conditionally expected count will be: $$\mathsf E(N(s)\mid N(t)=\alpha) ~=~ \dfrac {s\alpha}t$$

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The long way

Begin here:

$$\mathsf E(N(s)\mid N(t){=}\alpha) ~=~ \sum_{n=0}^\alpha n~\mathsf P(N(s){=}n\mid N(t){=}\alpha)$$

Now, do you know how to show?: $~\mathsf P(N(s){=}n\mid N(t){=}\alpha) ~=~ \dbinom{\alpha}{n}\dfrac{s^n(t-s)^{\alpha-n}}{t^\alpha}~\Big[n\in\{0,..,\alpha\}\Big]$

Then continue: