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

\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 X_i be the amount that the ith customer withdraws, and assume that X_1, X_2, ... is a sequence of i.i.d.r.v.'s independent of (N_t). The amount of money in the bank at time t is denoted by Y_t (a compound poisson process). We have

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
Pr (X_1 = \$1000 \times k ) = \frac{\exp(-1)}{k!} for k = 0,1,....