Question: Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters λi, Think of the N, as the number of Points of scatter in disjoint part of the plane with areas λi where the mean intensity is on point per unit area.
a) What is the conditional joint distribution of (Ni, , , ,, ,Nm) given Ni + . . .. +Nm = n?
b) Suppose now that N has Poisson (λ) distribution, and given N = n the conditional joint distribution of some m-tuple of random variables (Ni, . .. Nm) is exactly what you found in part a). What can you conclude about the unconditional distribution of (Ni.. .. Nm)?