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Chapter 9: Q. 9.7 (page 412)
A transition probability matrix is said to be doubly
stochastic if
for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then
j = 1/(M + 1), j = 0, 1, ... , M.
It is proved that a Markov chain is ergodic, then for
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Events occur according to a Poisson process with rate 位 = 3 per hour. (a) What is the probability that no events occur between times 8 and 10 in the morning? (b) What is the expected value of the number of events that occur between times 8 and 10 in the morning? (c) What is the expected time of occurrence of the fifth event after 2 P.M.?
Cars cross a certain point in the highway in accordance with a Poisson process with rate 位 = 3 per minute. If Al runs blindly across the highway, what is the probability that he will be uninjured if the amount of time that it takes him to cross the road is s seconds? (Assume that if he is on the highway when a car passes by, then he will be injured.) Do this exercise for s = 2, 5, 10, 20.
Let X be a random variable that takes on 5 possible values with respective probabilities .35, .2, .2, .2, and .05. Also, let Y be a random variable that takes on 5 possible values with respective probabilities .05, .35, .1, .15, and .35. (a) Show that H(X) > H(Y). (b) Using the result of Problem 9.13, give an intuitive explanation for the preceding inequality.
Compute the limiting probabilities for the model of Problem 9.4.
In transmitting a bit from location A to location B, if we let X denote the value of the bit sent at location A and Y denote the value received at location B, then H(X) 鈭� HY(X) is called the rate of transmission of information from A to B. The maximal rate of transmission, as a function of P{X = 1} = 1 鈭� P{X = 0}, is called the channel capacity. Show that for a binary symmetric channel with P{Y = 1|X = 1} = P{Y = 0|X = 0} = p, the channel capacity is attained by the rate of transmission of information when P{X = 1} = 1 2 and its value is 1 + p log p + (1 鈭� p)log(1 鈭� p).
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