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

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

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

for k = 0,1,....