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