1. The problem statement, all variables and given/known data There are two urns, A and B. Let v be a random number of balls. Each of these balls is put to urn A with probably p and to urn B with probability q = 1 - p. Let v_a and v_b denote the numbers of balls in A and B, respectively. Show that random variables v_a and v_b are independent if and only if v is a Poisson random variable. 2. Relevant equations The probability mass function of a Poisson random variable is 3. The attempt at a solution We showed in an earlier problem that if two variables X, Y are independent Poisson random variables then X + Y is also a Poisson random variable. Therefore, if v_A and v_B are Poisson random variables, then v must be as well. Noting that v_A + v_B = v, I think that proving that v must be a Poisson random variable might be sufficient, but I also don't know the level of rigor wanted here. It would be easy to show that v CAN be a Poisson random variable, but I don't see how to go about showing that it MUST be. I thought maybe I could show that the basis of "Poisson random variable space" is independent from the basis of "non-Poisson random variable space", but I don't know if those are even sensible ideas. Random thing I noted that may or may not be relevant: E(v) = E(v_A) + E(v_B). If v_A and v_B are Poisson random variables then their expectation value is their parameter \lambda.