This question is from a workbook i'm currently working on. If we have a poisson process thats is on a real line and denote it with $S(x_1,x_2)$ as the number of events in the time interval between $x_1$ and $x_2$. If we have $x_0<x_1<x_2$, what would be the conditional distribution of $S(x_0,x_1)$ knowing that $S(x_0,x_2)=c$?
The hint provided is to use the fact that the amount of events in disjoint subsets are independent.
I'm not sure how to go about doing this question. Ideally, I'd like an additional hint to work off of and perhaps a spoiler containing the solution but any and all help will be much appreciated!
Best Answer
Hint1: let us denote the Poisson arrival rate by $\lambda$. The expected number of events on an interval $x_1 < x_2$ is then $\lambda(x_2-x_1)$. What does this tell us about on the expected nb of events over one interval relativ to another? Does it provide a way to split the $c$ on these two intervals? The conditional distribution will not be Poisson (max nb events is limited), but what does independency give us? Is one interval more likely than another of same length to have a certain nb of events?
Hint2: Let $K$ be the random variable we are looking for. We know it can take values in $\{0,1,2,...,c\}$.
Hint3: Let $N_1$ be the Poisson distribution with parameter $\lambda(x_1-x_0)$ and $N_2$ Poisson with $\lambda(x_2-x_1)$. We have to find the distribution of $N_1$ given $N_1+N_2=c$. This sum of Poissons $N_1+N_2$ is also Poisson distributed, by independece (use for example the characteristic funtion or mgf). Now we have to find
$Prob(N_1=k, N_2=c-k | N_1+N_2=c)$
for $k \in \{0,1,...,c\}$. These probabilities describe the distribution of our $K$.
Hint 4: Bayes formula for conditional probabilities or Binomial distribution.