91Ó°ÊÓ

Teams 1 and 2 are playing a match. The teams score points according to independent Poisson processes with respective rates \(\bar{\lambda}_{1}\) and \(\lambda_{2}\). If the match ends when one of the teams has scored \(k\) more points than the other, find the probability that team 1 wins. Hint: Relate this to the gambler's ruin problem.

The water level of a certain reservoir is depleted at a constant rate of 1000 units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson process with rate \(.2\) per day. The amount of water added to the reservoir by a rainfall is 5000 units with probability \(.8\) or 8000 units with probability .2. The present water level is just slightly below 5000 units. (a) What is the probability the reservoir will be empty after five days? (b) What is the probability the reservoir will be empty sometime within the next ten days?

Events occur according to a Poisson process with rate \(\lambda=2\) per hour. (a) What is the probability that no event occurs between 8 P.M. and 9 P.M.? (b) Starting at noon, what is the expected time at which the fourth event occurs? (c) What is the. probability that two or more events occur between \(6 \mathrm{P.M}\). and 8 P.M.?

There are two types of claims that are made to an insurance company. Let \(N_{l}(t)\) denote the number of type \(i\) claims made by time \(t\), and suppose that \(\left[N_{1}(t), t \geqslant 0\right\\}\) and \(\left\\{N_{2}(t), t \geqslant 0\right]\) are independent Poisson processes with rates \(\lambda_{1}=.10\) and \(\lambda_{2}=1 .\) The amounts of successive type 1 claims are independent exponential random variables with mean \(\$ 1000\) whereas the amounts from type 2 'claims are independ?nt exponential random variables with mean \(\$ 5000\). A claim for \(\$ 4000\) has just been received; what is the probability it is a type 1 claim?

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate \(\lambda\). The bus departs at time \(t\). Let \(X\) denote the total amount of waiting time of all those who get on the bus at time \(t\). We want to determine \(\operatorname{Var}(X)\). Let \(N(t)\) denote the number of arrivals by time \(t .\) (a) What is \(E[X \mid N(t)] ?\) (b) Argue that \(\operatorname{Var}[X \mid N(t)]=N(t) t^{2} / 12\) (c) What is \(\operatorname{Var}(X) ?\)

An average of 500 people pass the California bar exam each year. A California lawyer practices law, on average, for 30 years. Assuming these numbers remain steady, how many lawyers would you expect California to have in \(2050 ?\)

A store opens at 8 A.M. From 8 until 10 customers arrive at a Poisson rate of four an hour. Between 10 and 12 they arrive at a Poisson rate of eight an hour. From 12 to 2 the arrival rate increases steadily from eight per hour at 12 to ten per hour at 2; and from 2 to 5 the arrival rate drops steadily from ten per hour at 2 to four per hour at \(5 .\) Determine the probability distribution of the number of customers that enter the store on a given day.

An insurance company pays out claims on its life insurance policies in accordance with a Poisson process having rate \(\lambda=5\) per week. If the amount of money paid on each policy is exponentially distributed with mean \(\$ 2000\), what is the mean and variance of the amount of money paid by the insurance company in a four-week span?

A two-dimensional Poisson process is a process of randomly occurring events in the plane such that (i) for any region of area \(A\) the number of events in that region has a Poisson distribution with mean \(\lambda A\) and (ii) the number of events in nonoverlapping regions are independent. For such a process, consider an arbitrary point in the plane and let \(X\) denote its distance from its nearest event (where distance is measured in the usual Euclidean manner). Show that (a) \(P\\{X>t\\}=e^{-\lambda \pi t^{2}}\) (b) \(E[X]=\frac{1}{2 \sqrt{\lambda}}\)