Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 8 / b / ii / Solution

For $0\le k\le n$, the event $\{X=k,X+Y=n\}$ is $\{X=k,Y=n-k\}$. Use independence in its numerator and the Poisson distribution of the sum in its denominator:Outside $0\le k\le n$ the conditional probability is zero. ConsequentlyThis binomial distribution includes $n=0$, when it is concentrated at zero. Conditioning on the total converts the independent counts into an allocation of that total between the two rates.