Question

I buy N lottery tickets where . Each ticket costs $1. For each ticket, I have probability of 1/50 of winning $10; otherwise I win nothing. You may assume independence of amounts won on each ticket.

(a) Let X be my profit from a single ticket. Write down the moment generating function of X.

(b) Les S be my total profit. Find an expression for the cumulant generating function ofS and hence evaluate E(S).

Answer.

(a) If X is the profit from a single ticket, then

Therefore

(b) Total profit S can be expressed as random sum

Therefore

and (Poisson)

Evaluate at t=0:

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