91Ó°ÊÓ

Let \(X\) be an exponential random variable. Without any computations, tell which one of the following is correct. Explain your answer. (a) \(E\left[X^{2} \mid X>1\right]=E\left[(X+1)^{2}\right]\) (b) \(E\left[X^{2} \mid X>1\right]=E\left[X^{2}\right]+1\) (c) \(E\left[X^{2} \mid X>1\right]=(1+E[X])^{2}\)

Consider a post office with two clerks. Three people, \(A, B\), and \(C\), enter simultaneously. A and B go directly to the clerks, and \(\mathrm{C}\) waits until either \(\mathrm{A}\) or \(\mathrm{B}\) leaves before he begins service. What is the probability that \(\mathrm{A}\) is still in the post office after the other two have left when (a) the service time for each clerk is exactly (nonrandom) ten minutes? (b) the service times are \(i\) with probability \(\frac{1}{3}, i=1,2,3 ?\) (c) the service times are exponential with mean \(1 / \mu\) ?

The lifetime of a radio is exponentially distributed with a mean of ten years. If Jones buys a ten-year-old radio, what is the probability that it will be working after an additional ten years?

Machine 1 is currently working. Machine 2 will be put in use at a time \(t\) from now. If the lifetime of machine \(i\) is exponential with rate \(\lambda_{i}, i=1,2\), what is the probability that machine 1 is the first machine to fail?

Let \(X\) be an exponential random variable with rate \(\lambda\). (a) Use the definition of conditional expectation to determine \(E[X \mid Xc] P\\{X>c\\} $$

Customers can be served by any of three servers, where the service times of server \(i\) are exponentially distributed with rate \(\mu_{i}, i=1,2,3\). Whenever a server becomes free, the customer who has been waiting the longest begins service with that server. (a) If you arrive to find all three servers busy and no one waiting, find the expected time until you depart the system. (b) If you arrive to find all three servers busy and one person waiting, find the expected time until you depart the system.

Let \(S(t)\) denote the price of a security at time \(t\). A popular model for the process \(\\{S(t), t \geqslant 0\\}\) supposes that the price remains unchanged until a "shock" occurs, at which time the price is multiplied by a random factor. If we let \(N(t)\) denote the number of shocks by time \(t\), and let \(X_{i}\) denote the \(i^{\text {th }}\) multiplicative factor, then this model supposes that $$ S(t)=S(0) \prod_{l=1}^{N(t)} X_{i} $$ where \(\prod_{l=1}^{N(t)} X_{i}\) i? equal to 1 when \(N(t)=0\). Suppose that the \(X_{l}\) are independent exponential random variables with rate \(\mu\); that \(\\{N(t), t \geqslant 0\\}\) is a Poisson process with rate \(\lambda\); that \([N(t), t \geqslant 0\\}\) is independent of the \(X_{i}\); and that \(S(0)=s\) (a) Find \(E[S(t)]\) (b) Find \(E\left[S^{2}(t)\right]\)

Cars pass 'a certain street location according to a Poisson process with ratee A woman who wants to cross the street at that location waits until she can see that no cars will come by in the next \(T\) time units. (a) Find the probability that her waiting time is \(0 .\) (b) Find her expected waiting time. Hint: Condition on the time of the first car.

Consider a two-server parallel queueing system where customers arrive according to a Poisson process with rate \(\lambda\), and where the service times are exponential with rate \(\mu\). Moreover, suppose that arrivals finding both servers busy immediately depart without receiving any service (such a customer is said to be lost), whereas those finding at least one free server immediately enter service and then depart when their service is completed. (a) If both servers are presently busy, find the expected time until the next customer enters the system. (b) Starting empty, find the expected time until both servers are busy. (c) Find the expected time between two successive lost customers.

The number of hours between successive train arrivals at the station is uniformly distributed on \((0,1)\). Passengers arrive according to a Poisson process with rate 7 per hour. Suppose a train has just left the station. Let \(X\) denote the number of people who get on the next train. Find (a) \(E[X]\) (b) \(\operatorname{Var}(X)\)