Question

I buy N lottery tickets where N{\sim}Pois(\lambda). 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

f_X(x)=P(X=x)=:

\frac{1}{50}, x=9

\frac{49}{50}, x=-1

0, otherwise

Therefore M_X(x)=E(e^{tX})=\Sigma_xe^{tx}f_X(x)=e^{9t}\frac{  1}{50}+e^{-t}\frac{49}{50}

(b) Total profit S can be expressed as random sum \Sigma_{i=1}^NX_i

Therefore K_S(t)=K_N(K_X(t))

K_X(t)=ln(M_X(t))=ln(1/50e^{9t}+49/50e^{-t}) and K_N(s)=\lambda(e^t-1) (Poisson)

K_S(t)=\lambda(exp^{ln(1/50e^{9t}+49/50e^{-1})}-1)=\frac{\lambda}{50}e^{9t}+\frac{49\lambda}{50}e^  {-t}-\lambda

\frac{d}{dt}K_S(t)=\frac{9\lambda}{50}e^{9t}-\frac{49\lambda}{50}e^{-t}

Evaluate at t=0: E(S)=\frac{-40{\lambda}}{50}

your feedback is welcome. thanks!