1. The dispatcher at a central fire station has observed that the time between calls is an exponential random variable with a mean of 32 minutes.
a) It is now noon and the most recent call came in at 11.35AM.What is the probability that the next call will arrive before 12.30PM?
b) What is the probability that there will be exactly two calls during the next hour?
2. Shocks occur to a system according to a Poisson process of rate λ. Suppose that the system survives each shock with probability 2 ε [0, 1], independently of other shocks.
What is the probability that the system is surviving at time t > 0?
3. Jack likes to go fishing. While waiting for the fishes to bite, he formulates the following model for the process : fishes bite according to a Poisson process with intensity 4 bites per hour. Biting fishes are caught independently, and on average only one in two times.
a) What is the probability that six fishes bite during the first two hours?
b) What is the probability that he fails to catch any fishes during the first two hours?
c) What is the probability that, during the first two hours, six fishes bite and two of these are caught?
4. Customers enter a store according to a Poisson process of rate λ = 5 per hour. Independently, each customer buys something with probability p = 0:8 and leaves without making a purchase with probability q = 1 p = 0:2. Each customer buying something will spend an amount of money uniformly distributed between $1 and $101 (independently of the purchases of the other customers). What are the mean and the standard deviation of the total amount of money spent by customers within any given 10-hour day?
5. Men and women enter a supermarket according to independent Poisson processes having respective rates of two and four per minute. It is noon and there are currently 10 customers in the supermarket. From now on, what is the probability that at least two men arrive before the second woman arrives?