A troubled miser stores his coins in a large number of boxes. Each box has a different number of coins; if you pick a box at random, the number of coins N in that box will have the Poisson distribution with parameter λ. That is, for all k ≥ 0,
P(N=k)=e−λλk/k!
Every coin has the same probability p of coming up Heads. If you pick a box at random and toss all the coins inside that box, calculate the probability that you get n Heads, for all n ≥ 0. [Hint: first condition on N].
P(N=k)=e−λλk/k!
Every coin has the same probability p of coming up Heads. If you pick a box at random and toss all the coins inside that box, calculate the probability that you get n Heads, for all n ≥ 0. [Hint: first condition on N].