Question

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

$\displaystyle f_X(x)=P(X=x)=:$

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

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

$\displaystyle 0, otherwise$

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

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

Therefore $\displaystyle K_S(t)=K_N(K_X(t))$

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

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

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

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

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