91Ó°ÊÓ

Customers arrive at a bank at a Poisson rate λ. Suppose that two customers arrived during the first hour. What is the probability that

(a) both arrived during the first 20 minutes?

(b) at least one arrived during the first 20 minutes?

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.

Determine the entropy of the sum that is obtained when a pair of fair dice is rolled.

Cars cross a certain point in the highway in accordance with a Poisson process with rate λ = 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.

Suppose that 3 white and 3 black balls are distributed in two urns in such a way that each urn contains 3 balls. We say that the system is in state i if the first urn contains i white balls, i = 0, 1, 2, 3. At each stage, 1 ball is drawn from each urn and the ball drawn from the first urn is placed in the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth stage, and compute the transition probabilities of the Markov chain {Xn, n Ú 0}.

A transition probability matrix is said to be doubly

stochastic if

$\underset{i=0}{\overset{M}{∑}}{P}_{ij}=1$

for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then

j = 1/(M + 1), j = 0, 1, ... , M.

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?