Page 215 - Probability, Random Variables and Random Processes

CHAP. 51 RANDOM PROCESSES

(a) Find the cross-correlation function of Rxy(t, t + z) of X(t) and Y(t).
(b) Repeat (a) if 63 = 4.
Ans. (a) Rxy(t, t + z)] = 0
A04
(b) Rxy(t, t + 2) =- cos[(ol - u,)t + o,z]
2
Given a Markov chain {X, , n 2 01, find the joint pmf
P(X, = i,, X1 = i,, ..., X, = in)
Hint: Use Eq. (5.32).

Let {X,, n 2 0) be a homogeneous Markov chain. Show that
P(Xn+,=kl ,..., X,+,=k,IX,=i ,,..., X,=i)=P(X, =k, ,..., X,=k,IX,=i)
Hint: Use the Markov property (5.27) and the homogeneity property.

Verify Eq. (5.37).
Hint: Write Eq. (5.39) in terms of components.

Find Pn for the following transition probability matrices:

A certain product is made by two companies, A and B, that control the entire market. Currently, A and B
have 60 percent and 40 percent, respectively, of the total market. Each year, A loses 5 of its market share to
By while B loses 3 of its share to A. Find the relative proportion of the market that each hold after 2 years.
Ans. A has 43.3 percent and B has 56.7 percent.

Consider a Markov chain with state (0, 1, 2) and transition probability matrix

Is state 0 periodic?
Hint: Draw the state transition diagram.
Ans. No.

5.74. Verify Eq. (5.51).