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The question is:

" Let $ \{N(t) , t \geq 0 \} $ be a $PP(\lambda)$. Compute

$$P(N(t) = k | N(t = s) = k + m), $$

where k and m are non-negative integers and $ t, s \geq 0 $ are any real numbers".

So I did this:

Using the definition of joint probability, you get:

$$ P(N(t) = k | N(t = s) = k + m) = \frac{P(N(t) = k, N(t + s) = k + m)}{P(N(t + s) = k + m)} $$

Then I got a bit stuck, looking at the answersm, this happens:

From what I did, it goes to:

$$P(N(t) = k | N(t = s) = k + m) = \frac{P(N(t) = k, N(t + s) - N(t) =m)}{P(N(t + s) = k + m)} $$

Because a PP has independent increments:

$$= \frac{ \left( e^{ \lambda t} \frac{ (\lambda t)^k}{k!} \right) \cdot \left( e^{\lambda s} \frac{(\lambda s)^m}{m!} \right) } { \left( e^{ \lambda (t + s)} \frac{(\lambda (t + s))^{(k+m)}}{(k+m)!} \right) } $$

Because the increments have a Poisson distribution

$$= \begin{pmatrix} k + m & k \end{pmatrix} \frac{t^k s^m}{(t + s)^{k+m} } $$

The questions I have are these:

1) Why do you go from $ N(t + s) = k + m $to $ N(t + s) - N(t) = m$? Does it make any difference if you don't do this?

2) Why do you ignore the -N(t) when writing it in its "exponential form"?

3) What does the last nCr bit mean?

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1) Think of each of those $k+m$ events as having occurred either before $t$ or after $t$, and you're trying to calculate the probability that exactly $k$ of them occurred before $t$, which leaves exactly $m$ of them occurring after $t$.

2) $N(t+s)-N(t)$ is just the number of events occurring during a period of length $s$; since this is a Poisson process with constant rate, the probability that $m$ occur in that interval is independent of $N(t)$, depending solely upon $s$, so we can just refer to $s$ instead.

3) nCr is a shorthand for $n$ choose $r$, often written $n \choose r$.

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