1. Telephone calls arrive at a switchboard in a Poisson process at the rate of 2 per minute. A random one-tenth of the calls are long distance.

(a) What is the probability of at least one long distance call in a ten minute period?

(b) Given that there have been 8 long distance calls in an hour, what is the expected number of calls to have arrived in the same period?

(c) Given that there were 90 calls in an hour, what is the probability that 10 were long distance?

2. The number of missing items in a certain location, call it X, is a Poisson random variable with mean λ. When searching the location, each item will independently be found after an independently distributed time with rate µ. A reward of R is received for each item found, and a searching cost of C per unit of search time is incurred. Suppose that you search for a ?xed time t and then stop.

(a) Find your total expected return.

(b) Find the value of t that maximizes the total expected return.

3. Due to the stress of coping with assignments you begin to experience migraine headaches of random severities. The occurrence of headaches follows a Poisson Process with rate λ. Headache severities are independent of times of occurrences and are iid exponentially distributed random variables with rate µ. You decide to commit yourself to the hospital if a headache of severity greater than c > 0 occurs.

Compute the probability that you do not commit yourself to the hospital in [0, t].

4. Consider an in?nite server queueing system where jobs arrive according to a Poisson process with rate λ. The service times of the jobs are iid random variables with cumulative distribution function G(y). For some t > c ≥ 0, ?nd the expected number of jobs that arrived by time c and not completed their service by time t.