Chapter 9: adasdasdsad (page 395)
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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.
A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X equal to either x1 or x2 or x3?” and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?
Consider Example 2a. If there is a 50–50 chance of rain today, compute the probability that it will rain 3 days from now if α = .7 and β = .3.
A certain person goes for a run each morning. When he leaves his house for his run, he is equally likely to go out either the front or the back door, and similarly, when he returns, he is equally likely to go to either the front or the back door. The runner owns 5 pairs of running shoes, which he takes off after the run at whichever door he happens to be. If there are no shoes at the door from which he leaves to go running, he runs barefooted. We are interested in determining the proportion of time that he runs barefooted. (a) Set this problem up as a Markov chain. Give the states and the transition probabilities. (b) Determine the proportion of days that he runs barefooted.
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) − 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 − 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 − p)log(1 − p).
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