The Poisson random variable is often used to describe the number of occurrences of a phenomenon (or events) per unit of time. For example, the number of requests to an internet service per second, the number of earthquake, in a thousand year. the number of customers wining in a shop in one hour, all can be appropriately modeled by a Poisson distribution, as long as the rate (or the average number) of events (earthquakes, requests, customers coming) are constant. The exponential distribution, on the other hand, measures the time elapsed between two consecutive events. This problem will roughly explain this relation between Passion and exponential distributions.

Assume that events occur according to a Poisson distribution with a constant rate λ (events per unit of time).

(a) Find the probability mass function (PMF) of the random variable X, that measures the number of events occurring in t units of time.

(b) Let T measure the waiting time from the 'beginning" to the first event (the beginning of time can be set to any time, in particular. if we set it to be the time one event occurs, then T measures time elapsed between the two events). Find the cumulative distribution function of T, F _T (t)=Pr(T ≤ t), based on the PMF of X _l and conclude that T follows an exponential distribution with the same parameter λ .

(c) Suppose that on average, 30 customers arrive at a shop per hour, and that the number of customers follows a Poisson distribution. Find the probability that the shopkeeper will have to wait more than 10 minutes before the first customer arrives?

(d) Continuing on part (c), what is the expected duration of time between any two consecutive customers' arrivals? (Do you find your answer intuitively clear?)