91Ó°ÊÓ

There are two types of claims that are made to an insurance company. Let \(N_{i}(t)\) 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 independent exponential random variables with mean \(\$ 5000 .\) A claim for \(\$ 4000\) has just been received; what is the probability it is a type 1 claim?

Customers arrive at a bank at a Poisson rate \(\lambda .\) Suppose 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?

Policyholders of a certain insurance company have accidents at times distributed according to a Poisson process with rate \(\lambda .\) The amount of time from when the accident occurs until a claim is made has distribution \(G\). (a) Find the probability there are exactly \(n\) incurred but as yet unreported claims at time \(t\). (b) Suppose that each claim amount has distribution \(F\), and that the claim amount is independent of the time that it takes to report the claim. Find the expected value of the sum of all incurred but as yet unreported claims at time \(t\).

Shocks occur according to a Poisson process with rate \(\lambda\), and each shock independently causes a certain system to fail with probability \(p .\) Let \(T\) denote the time at which the system fails and let \(N\) denote the number of shocks that it takes. (a) Find the conditional distribution of \(T\) given that \(N=n\). (b) Calculate the conditional distribution of \(N\), given that \(T=t\), and notice that it is distributed as 1 plus a Poisson random variable with mean \(\lambda(1-p) t .\) (c) Explain how the result in part (b) could have been obtained without any calculations.

Suppose that the times between successive arrivals of customers at a single- server station are independent random variables having a common distribution \(F .\) Suppose that when a customer arrives, he or she either immediately enters service if the server is free or else joins the end of the waiting line if the server is busy with another customer. When the server completes work on a customer, that customer leaves the system and the next waiting customer, if there are any, enters service. Let \(X_{n}\) denote the number of customers in the system immediately before the \(n\) th arrival, and let \(Y_{n}\) denote the number of customers that remain in the system when the \(n\) th customer departs. The successive service times of customers are independent random variables (which are also independent of the interarrival times) having a common distribution \(G\). (a) If \(F\) is the exponential distribution with rate \(\lambda\), which, if any, of the processes \(\left\\{X_{n}\right\\},\left[Y_{n}\right\\}\) is a Markov chain? (b) If \(G\) is the exponential distribution with rate \(\mu\), which, if any, of the processes \(\left\\{X_{n}\right\\},\left\\{Y_{n}\right\\}\) is a Markov chain? (c) Give the transition probabilities of any Markov chains in parts (a) and (b).

Suppose that customers arrive to a system according to a Poisson process with rate \(\lambda\). There are an infinite number of servers in this system so a customer begins service upon arrival. The service times of the arrivals are independent exponential random variables with rate \(\mu\), and are independent of the arrival process. Customers depart the system when their service ends. Let \(N\) be the number of arrivals before the first departure. (a) Find \(P(N=1)\). (b) Find \(P(N=2)\) (c) Find \(P(N=j)\). (d) Find the probability that the first to arrive is the first to depart. (e) Find the expected time of the first departure.

Let \(T_{1}, T_{2}, \ldots\) denote the interarrival times of events of a nonhomogeneous Poisson process having intensity function \(\lambda(t)\). (a) Are the \(T_{i}\) independent? (b) Are the \(T_{i}\) identically distributed? (c) Find the distribution of \(T_{1}\).

Let \(X_{1}, X_{2}, \ldots\) be independent positive continuous random variables with a common density function \(f\), and suppose this sequence is independent of \(N, a\) Poisson random variable with mean \(\lambda\). Define $$ N(t)=\text { number of } i \leqslant N: X_{i} \leqslant t $$ Show that \(\\{N(t), t \geqslant 0\\}\) is a nonhomogeneous Poisson process with intensity function \(\lambda(t)=\lambda f(t)\).

In good years, storms occur according to a Poisson process with rate 3 per unit time, while in other years they occur according to a Poisson process with rate 5 per unit time. Suppose next year will be a good year with probability \(0.3\). Let \(N(t)\) denote the number of storms during the first \(t\) time units of next year. (a) Find \(P\\{N(t)=n]\). (b) Is \([N(t)\\}\) a Poisson process? (c) Does \(\\{N(t)\\}\) have stationary increments? Why or why not? (d) Does it have independent increments? Why or why not? (e) If next year starts off with three storms by time \(t=1\), what is the conditional probability it is a good year?

Determine \(\operatorname{Cov}[X(t), X(t+s)]\) when \(\\{X(t), t \geqslant 0\\}\) is a compound Poisson process.