statistics
Notation
In this assignment, Poisson process with intensity λ > 0 is denoted as
X = {X(t) : t ≥ 0}
Arrival
times are defined for any natural k ≥ 1 as follows:
Wk = min [t ≥ 0 : X(t) = k]
and W0 = 0 by definition. The interpretation is that Wk is the time of k
th arrival. Sojourn (or inter-arrival)
times are defined for any natural k ≥ 1 as follows:
Sk = Wk −Wk−1 ⇐⇒ Wk =
k∑
j=1
Sj
Facts about Poisson processes and related distributions were presented in the Notes, Part 5. You can (and
should) use them and other materials from the folder titled Poisson.
2
Problem 1 [10 points]
Customer arrivals at a service center form a Poisson process with intensity λ = 12 per hour (or one per
5 minutes). Given that the first customer arrived three minutes after the center was open, find the
conditional expectation of arrival time for a third customer.
Solution
3
Problem 3 [10 points = 5 + 5]
Customer arrivals at a service center form a Poisson process with intensity λ = 12 per hour (or one per
5
minutes).
1. Evaluate expectation of the ratio,
Q =
W4
W5
2. Find expected value for
T =
W5 −W4
W4
Solution
5
Problem 4 [15 points = 5 + 5 + 5]
Customer arrivals at a service center form a Poisson process with intensity λ = 12 per hour (or one per 5
minutes).
1. Evaluate expectation for the ratio of two arrival times,
Q =
W3
W5
2. Find conditional expectation of W3, given that the fifth customer’s arrival was at t = 15.
3. Find expected value of the ratio,
T =
W5
W3
Solution
6