91Ó°ÊÓ

Mr. Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate does \(\mathrm{Mr}\). Smith get new jobs?

Events occur according to a Poisson process with rate \(\lambda\). Any event that occurs within a time \(d\) of the event that immediately preceded it is called a \(d\) -event. For instance, if \(d=1\) and events occur at times \(2,2.8,4,6,6.6, \ldots\), then the events at times \(2.8\) and \(6.6\) would be \(d\) -events. (a) At what rate do \(d\) -events occur? (b) What proportion of all events are \(d\) -events?

Consider the gambler's ruin problem where on each bet the gambler either wins 1 with probability \(p\) or loses 1 with probability \(1-p\). The gambler will continue to play until his winnings are either \(N-i\) or \(-i\). (That is, starting with \(i\) the gambler will quit when his fortune reaches either \(N\) or \(0 .\) ) Let \(T\) denote the number of bets made before the gambler stops. Use Wald's equation, along with the known probability that the gambler's final winnings are \(N-i\), to find \(E[T]\). Hint: Let \(X_{j}\) be the gambler's winnings on bet \(j, j \geqslant 1\). What are the possible values of \(\sum_{j=1}^{T} X_{j} ?\) What is \(E\left[\sum_{j=1}^{T} X_{j}\right]\) ?

A system consists of two independent machines that each function for an exponential time with rate \(\lambda .\) There is a single repairperson. If the repairperson is idle when a machine fails, then repair immediately begins on that machine; if the repairperson is busy when a machine fails, then that machine must wait until the other machine has been repaired. All repair times are independent with distribution function \(G\) and, once repaired, a machine is as good as new. What proportion of time is the repairperson idle?

Each time a certain machine breaks down it is replaced by a new one of the same type. In the long run, what percentage of time is the machine in use less than one year old if the life distribution of a machine is (a) uniformly distributed over \((0,2)\) ? (b) exponentially distributed with mean \(1 ?\)