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I could use some help with a homework question. Let ${N(t), t \geq 0}$ be a Poisson Process with rate $\lambda$. For $i\leq n $ and $s<t$, find: $P(N(t)=n|N(s)=i)$ and $P(N(s)=i|N(t)=n)$.

For the first part here is what I did:

I assumed the times are memoryless: $$P(N(t)=n|N(s)=i) = P(N(t-s)=n-i)$$

Then, we get $$ = \frac{e^{-\lambda(t-s)}(\lambda(t-s))^{n-i}}{(n-i)!}$$

So, I'm wondering if I have the correct approach. If not, please provide any suggestions.

The Mathemusician
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  • Please add what you have done to try to solve the question otherwise your question will be downvoted and closed. – Shubham Johri Oct 26 '20 at 16:47
  • Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. – saulspatz Oct 26 '20 at 16:53
  • Thanks for the suggestion. I just added part of my possible solution. – The Mathemusician Oct 26 '20 at 17:09
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    That's correct. To do the second half, start by writing down the formula for conditional probability. – saulspatz Oct 26 '20 at 17:24
  • See here for a hint for the second part: https://math.stackexchange.com/questions/151272/proof-that-conditional-of-poisson-distribution-is-binomial – Math1000 Oct 28 '20 at 05:35

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