How to prove this binomial distribution

Question

Assume that X1 ∼ P(µ1) and X2 ∼ P(µ2) are independent. Prove that the conditional distribution of X1 under the condition X1 + X2 = N to be the binomial distribution B(N, p),where p = µ1/(µ1 + µ2). How to do it ? I tried moment generating function but how to transform it into a binomial form?

Answer 1

Guide:

• First prove that $X_1+X_2\sim\mathsf{Poisson}(\mu_1+\mu_2)$ e.g. by finding an expression for $P(X_1+X_2=k)=\sum_{i=0}^kP(X_1=i\wedge X_2=k-i)$
• Now find an expression for $P(X_1=k\mid X_1+X_2=N)=\frac{P(X_1=k\wedge X_2=N-k)}{P(X_1+X_2=N)}$