Let N ∼ Poisson(λ). You flip a coin a random number of times N. Each time a head will come with probability p, independently of N. Let X be the (random) number of heads outcomes and Y be the (also random) number of tails. Find the distribution of X and Y. Are they independent? Give conditional distribution of N if X = k.
My idea is to condition on value of N and surely the number of heads of a coin toss is a binomial RV. So it's posisble to sum up the conditional probabilities of all values of N, i believe. X and Y are surely independent but I have trouble concluding this (since poisson distributions are uncorrelated) Any hints would be appreciated for second or third one. Thanks in advance.