Poisson proces and binomial

JesperHP

TS Contributor
I'm trying to find the distribution of the number $$Y$$ of arrivals in the
timeinterval $$[s,t]\subset [0,T]$$ where X(t) is number of arrivals in a poissonproces and it is given that X(T)=N.

The result I get is that:

$$Y=X(t-s)|X(T)=N$$ is binomial $$bin(n,p)$$ with n=N and p=|t-s|/T

where $$s < t$$ and $$[s,t]\subset [0,T]$$

I'm wondering if anyone would happen to know whether this result is correct?