Question

In each problem, make sure that you are clearly defining random variables, stating their distributions, and writing down the formulas that you are using. (That is, write down the pmf, write down mean and variance formulas.)

Customers arrive to the checkout counter of a convenience store according to a Poisson process at a rate of two per minute. Find the mean, variance, and the probability density function of the waiting time between the opening of the counter and the following events:

a. The arrival of the second customer.

b. The arrival of the third customer.

c. What is the probability that the third customer arrives within 6 minutes? You can use a computer if you’d like but you need to write down the integral with all of the numbers plugged in.

Answer 1

For a poisson process, the arrival times follow a gamma distribution with shape parameter = customer n, and rate parameter = average arrival rate, .

i.e.,

Therefore, expected arrival time of nth customer is given by , and variance is

a: Expected arrival time of second customer, E(X1 + X2) = 2/2 = 1 minute

Variance = 2/4 = 0.5

PDF of waiting time between the opening of the counter and arrival of the second customer is F(X) =

b: E(X1 + X2 + X3) = 3/2 = 1.5 minutes

Variance = 3/4 = 0.75

PDF =

c: P(X1 + X2 + X3 < 6) = P(Gamma(4,2) < 6) = =   = 0.1911