Let $X= (X(t); t\ge0)$ be a poisson process with the intensity ($\lambda$ per hour)
A) Find the conditional probability of having $m$ events in the first $t$ hours, given that there were $n$ events in first $T$ hours. (here $0\le m \le n$ and $0<t<T$)
I need someone to check if this is right.
$P(m$ events in $t$ hours | $n$ events in $T$ hours) = $P(m$ events in $t$ hours)/$P(n$ events in $T$ hours) =
$$ {((e^{-\lambda t})(\lambda t)^m) }/m! \over {((e^{-\lambda T})(\lambda T)^n) /n!} $$
Any help would be greatly appreciated.
Best Answer
Hint: In order for you answer to be correct you also need to add the following term in the numerator $$\begin{align*}&\phantom{\,\=}P(m \text{ events in $t$ hours | $n$ events in $T$ hours})= \\ \\& =\frac{P(m \text{ events in $t$ hours, $n-m$ events in $T-t$ hours})}{P(n \text{ events in $T$ hours})}=\\\\& =\frac{P(m \text{ events in $t$ hours})\cdot P(n-m \text{ events in $T-t$ hours})}{P(n \text{ events in $T$ hours})}=\ldots\end{align*}$$ Then the formula you used for the poisson distribution is correct. The result should reflect the binomial distribution (see second bullet).