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Chapter 9: Q. 9.12 (page 412)
Determine the entropy of the sum that is obtained when a pair of fair dice is rolled.
The entropy of the sum is when a pair of fair dice is rolled.
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Compute the limiting probabilities for the model of Problem 9.4.
A pair of fair dice is rolled. Let
and let Y equal the value of the first die. Compute (a) H(Y), (b) HY(X), and (c) H(X, Y).
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.
Suppose that whether it rains tomorrow depends on past weather conditions only through the past 2 days. Specifically, suppose that if it has rained yesterday and today, then it will rain tomorrow with probability .8; if it rained yesterday but not today, then it will rain tomorrow with probability .3; if it rained today but not yesterday, then it will rain tomorrow with probability .4; and if it has not rained either yesterday or today, then it will rain tomorrow with probability .2. What proportion of days does it rain?
Suppose that in Problem 9.2, Al is agile enough to escape from a single car, but if he encounters two or more cars while attempting to cross the road, then he is injured. What is the probability that he will be unhurt if it takes him s seconds to cross? Do this exercise for s = 5, 10, 20, 30.
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