Problem 1
Let \(\left\\{X_{1}(t), t \geq 0\right\\}\) and \(\left\\{X_{2}(t), t \geq 0\right\\}\) be independent Poisson processes with common intensity, \(\lambda\). Suppose that \(X_{1}(0)=9\) and \(X_{2}(0)=5 .\) What is the probability that the \(X_{1}\)-process reaches level 10 before the \(X_{2}\)-process does?
Problem 4
Consider a Poisson process with intensity \(\lambda\). We start observing at time \(t=0\). Let \(T\) be the time that has elapsed at the first occurrence. Continue to observe the process \(T\) further units of time. Let \(N(T)\) be the number of occurrences during the latter period (i.e., during \((T, 2 T])\). Determine the distribution of \(N(T)\).
Problem 5
A particle source, \(A\), emits one particle at a time, according to a Poisson process with an intensity of two particles a minute. Another particle source, \(B\), emits two particles at a time, according to a Poisson process with an intensity of one pair of particles a minute. The sources are independent of each other. We begin to observe the sources at time 0 . Compute the probability that source \(A\) has emitted two particles before source \(B\) has done so.
Problem 7
Sven waits for the bus. The waiting time, \(T\), until a bus comes is \(U(0, a)\)-distributed. While he waits he tries to get a ride from cars that pass by according to a Poisson process with intensity \(\lambda\). The probability of a passing car picking him up is \(p\). Determine the probability that Sven is picked up by some car before the bus arrives. Remark. All necessary independence assumptions are permitted.
Problem 10
A radio amateur wishes to transmit a message. The frequency on which she sends the Morse signals is subject to random disturbances according to a Poisson process with intensity \(\lambda\) per second. In order to succeed with the transmission, she needs a time period of \(a\) seconds without disturbances. She stops as soon as she is done. Let \(T\) be the total time required to finish. Determine \(E T\).
Problem 12
People arrive at an automatic transaction machine (ATM) according to a Poisson process with intensity \(\lambda\). The service time required at the ATM is constant, \(a\) seconds. Unfortunately, this machine does not allow for any waiting customers (i.e., no queue is allowed), which means that persons who arrive while the ATM is busy have to leave. When the \(a\) seconds of a customer have elapsed, the ATM is free to serve again, and so on. Suppose that the ATM is free at time 0 , and let \(T_{n}\) be the time of the arrival of the \(n\)th customer. Find the distribution of \(T_{n}\), and compute \(E T_{n}\) and \(\operatorname{Var} T_{n}\). Remark. Customers arriving (and leaving) while the ATM is busy thus do not affect the service time.
Problem 13
Customers arrive at a computer center at time points generated by a Poisson process with intensity \(\lambda\). The number of jobs brought to the center by the customers are independent random variables whose common generating function is \(g(u)\). Compute the generating function of the number of jobs brought to the computer center during the time interval \((s, t]\).
Problem 15
A Poisson process is observed during \(n\) days. The intensity is, however, not constant, but varies randomly day by day, so that we may consider the intensities during the \(n\) days as \(n\) independent, \(\operatorname{Exp}\left(\frac{1}{\alpha}\right)\) distributed random variables. Determine the distribution of the total number of occurrences during the \(n\) days.
Problem 17
Consider a queueing system, where customers arrive according to a Poisson process with intensity \(\lambda\) customers per minute. Let \(X(t)\) be the total number of customers that arrive during \((0, t]\). Compute the correlation coefficient of \(X(t)\) and \(X(t+s)\).
Problem 18
A particle is subject to hits at time points generated by a Poisson process with intensity \(\lambda\). Every hit moves the particle a horizontal, \(N\left(0, \sigma^{2}\right)\)-distributed distance. The displacements are independent random variables, which, in addition, are independent of the Poisson process. Let \(S_{t}\) be the location of the particle at time \(t\) (we begin at time 0 ). (a) Compute \(E S_{t}\). (b) Compute \(\operatorname{Var}\left(S_{t}\right)\). (c) Show that $$ \frac{S_{t}-E S_{t}}{\sqrt{\operatorname{Var}\left(S_{t}\right)}} \stackrel{d}{\longrightarrow} N\left(0, a^{2}\right) \quad \text { as } \quad t \rightarrow \infty $$ and determine the value of the constant \(a\).
