Page 216 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

Hint: Let = [Njk], where Njk is the number of times the state k(~ is occupied until absorption takes
B)
place when X(n) starts in state j(~ B). Then 7;. = ~~=,+, Njk; calculate E(Njk).
5.75. Consider a Markov chain with transition probability matrix

Find the steady-state probabilities.
Ans. p = [$ $ $1
5.76. Let X(t) be a Poisson process with rate A. Find E[X2(t)].
Ans. At + A2t2

5.77. Let X(t) be a Poisson process with rate 1. Find E([X(t) - X(s)I2) for t > s.
Hint: Use the independent stationary increments condition and the result of Prob. 5.76.
Ans. A(t - s) + A2(t - s)~

5.78. Let X(t) be a Poisson process with rate A. Find
P[X(t -d)= kIX(t)=j] d >O

j! (tid)k(:$-k
Ans.
k!(j - k)!
5.79. Let T, denote the time of the nth event of a Poisson process with rate A. Find the variance of T,.
Ans. n/A2

5.80. Assume that customers arrive at a bank in accordance with a Poisson process with rate 1 = 6 per hour, and
suppose that each customer is a man with probability 4 and a woman with probability 5. Now suppose
that 10 men arrived in the first 2 hours. How many woman would you expect to have arrived in the first 2
hours?
Ans. 4
5.81. Let X,, . . . , X, be jointly normal r.v.'s. Let
+
5 =Xi ci i = 1, ..., n
where ci are constants. Show that Y,, . . . , Y,, are also jointly normal r.v.'s.
Hint: See Prob. 5.60.

5.82. Derive Eq. (5.63).
Hint: Use condition (1) of a Wiener process and Eq. (5.1 02) of Prob. 5.22.