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Chapter 9: adasdasdsad (page 395)
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Show that for any discrete random variable and function
Prove that if X can take on any of n possible values with respective probabilities P1, ... ,Pn, then H(X) is maximized when Pi = 1/n, i = 1, ... , n. What is H(X) equal to in this case?
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?
Customers arrive at a certain retail establishment according to a Poisson process with rate 位 per hour. Suppose that two customers arrive during the first hour. Find the probability that
(a) both arrived in the first 20 minutes;
(b) at least one arrived in the first 30 minutes.
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.
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