statistics

Problem 3 [15 = 5 + 5 + 5 points]

Consider a birth and death process, X = {X(t) : t ≥ 0} with constant instant rates, λ and µ = 5 per hour, respectively.

• The process start at the state zero.

• Expected first arrival has E [S |X(0) = 0] = 15 minutes.

1. Derive the limiting distribution,

πk = lim t−→∞

P [X(t) = k] for all k ≥ 0.

2. Evaluate the limiting expectation, lim

t−→∞ E [X(t)]

3. Given that the process is initially at state k = 1, determine the expected departure time in minutes, that is:

E [S1 X(0) = 1]

Solution

4

Problem 5 [20 = 5 + 5 + 5 + 5 points]

Random variables, {Sk : k ≥ 1} are independent with common exponential distribution having

E [Sk] = 15 minutes

Introduce two variables as follows:

V = 4∑

k=1

Sk and W = 12∑ k=1

Sk

Use properties of a Poisson process to answer questions below.

1. Find expectation of a ratio,

Q = V

W

2. Derive expected value for a ratio,

T = W −V V

3. Evaluate conditional expectation (in hours) for V, given that W = 6 hours

4. Determine the expected value of W (in hours), given that V = 2 hours.

Solution

6

Problem 6 [10 points = 5 + 5]

Consider a discrete time Markov chain that describes the changes in brand name preferences among a large group of customers. There are THREE brand names, say A, B, and C. The transitions are described as follows.

A B C

A 0.4 0.5 0.1

B 0.6 0.4 0

C 0 0.2 0.8

1. Find transition probabilities after 2 time units and then express the probability,

P [X(n + 2) = A |X(n) = A]

2. In a long run, find proportions of customers’ preferences as a stationary distribution,

q = (qA, qB, qC )

Solution

7