## Compound Poisson Process

Suppose that customers arrive at a particular bank at the times of a Poisson process $\displaystyle (N_t)$ with time-dependent rate function

$\displaystyle \lambda(t)=\exp(t), \ \ 0 \leq t \leq 8$

where t is measure in hrs after the opening time of 8:30am. The bank closes its doors to customers at 4:30pm. Let $\displaystyle X_i$ be the amount that the ith customer withdraws, and assume that $\displaystyle X_1, X_2, ...$ is a sequence of i.i.d.r.v.'s independent of $\displaystyle (N_t)$. The amount of money in the bank at time t is denoted by $\displaystyle Y_t$ (a compound poisson process). We have

$\displaystyle Y_t = Y_0 - X_0 - X_1 - X_2 - ... - X_{N_t}$

What is the probability that a total of 3000 customers arrive during business hours, with exactly 30 arriving before noon?

Now, if the bank initially has 3 million dollars cash, calculate the expected amount of cash the bank has after today's bank run if
$\displaystyle Pr (X_1 = \$1000 \displaystyle \times k ) = \frac{\exp(-1)}{k!}\$ for k = 0,1,....