Suppose N 1 ,...,N m are independent Poisson processes on a space S with respective intensities μ 1 ,...,μ m , and let N = m i=1 N i , which is a Poisson process with intensity μ = μ 1 + ··· μ m . For...

Suppose N₁,...,N_m
are independent Poisson processes on a space S with respective intensities μ₁,...,μ_m, and let N = m i=1 N_i, which is a Poisson process with intensity μ = μ₁
+ ··· μ_m. For instance, N_i(B) might be the number of crimes of type i in a region B of a city and N(B) is the total number of crimes. Show that the conditional distribution of N₁(B),...,N_m(B) given N(B) = n is a multinomial distribution.