91Ó°ÊÓ

Events occur according to a Poisson process with rate \(\lambda=3\) 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 occurrence time of the fifth event after 2 P.M?

Cars cross a certain point in the highway in accordance with a Poisson process with rate \(\lambda=3\) per minute. If \(A l\) blindly runs across the highway, then 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 it for \(s=2,5,10,20\).

Four out of every five trucks on the road are followed by a car, while one out of every six cars is followed by a truck. What proportion of vehicles on the road are trucks?

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 back door. The runner owns 5 pairs of running shoes which he takes off after the run at whichèver 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 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 obtained when a pair of fair dice is rolled.

A pair of fair dice is rolled. Let $$ X= \begin{cases}1 & \text { if the sum of the dice is } 6 \\ 0 & \text { otherwise }\end{cases} $$ and let \(Y\) equal the value of the first die. Compute (a) \(H(Y)\), (b) \(H_{Y}(X)\), and (c) \(H(X, Y)\).

For any discrete random variable \(X\) and function \(f\), show that $$ H(f(X)) \leq H(X) $$