Math Problems Vault

Problem 33.1

(a)(b)(c)

Let $\xi (t)$ be a Poisson process with parameter $\lambda$ . Suppose each event is registered'' with probability $p$, independently of the other events. Let $\eta (t)$ be the process whose jumps occur only at the times of the registered'' events. Prove that

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(a)

$\eta (t)$ is a Poisson process with parameter $p\lambda$ ;

(b)

$\xi (t) - \eta (t)$ is a Poisson process with parameter $(1-p)\lambda$ ;

(c)

the processes $\eta (t)$ and $\xi (t) - \eta (t)$ are mutually independent.

Remark.

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A stochastically continuous homogeneous process with independent increments $\xi (t)$ is called a Poisson process if $\xi (0) = 0$ and $\xi (1)$ has a Poisson distribution. Throughout the rest of this section, modifications of the Poisson process having step sample paths with unit jumps are considered.

Problem 33.2

(a)(b)

Let $\xi_{1}(t)$ and $\xi_{2}(t)$ be two independent Poisson processes with parameters $\lambda_{1}$ and $\lambda_{2}$ . Prove that

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(a)

the process $\xi (t) = \xi_{1}(t) + \xi_{2}(t)$ is Poisson with parameter $\lambda = \lambda_{1} + \lambda_{2}$ ;

(b)

each point of the process $\xi (t)$ belongs to the process $\xi_{1}(t)$ with probability $p = \lambda_{1}/\lambda$ , independently of the other points.

Problem 33.3

Let $(\xi (t), \eta (t))$ be a two-dimensional random process, where $\xi (t)$ is a Poisson process with parameter $\lambda$ and $\eta (t)$ is a Poisson process with parameter $\mu$ , independent of $\xi (t)$ . Given that the process is in state $(x_{0}, y_{0})$ at time $t=0$ , with $x_{0}+y_{0} < z$ , find the probability that the process crosses the line $x'+y'=z$ at the point $(x, y)$ of this line.

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Problem 33.4

(a)(b)(c)

Consider a Poisson process with variable intensity, i.e., the probability of a jump occurring in the time interval $(t, t+h)$ does not depend on the past and equals $\lambda (t)h + o(h)$ as $h \to 0$ (note that $\lambda$ may depend on $t$ ).

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(a)

Prove that the probability of no jumps on the time interval $[0, t]$ equals

\exp \left\{ -\int _{0}^{t} \lambda (s)\, ds\right\} .

(b)

Let $x_{1}$ be the first point of the process. Find its distribution.

(c)

Prove that the probability of exactly $k$ jumps on the time interval $[0, t]$ equals

\frac{1}{k!}\left(\int _{0}^{t} \lambda (s)\, ds\right)^{k} \exp \left\{ -\int _{0}^{t} \lambda (s)\, ds\right\} .

Problem 33.5

(a)(b)

Find the covariance function of a Poisson process:

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(a)

with constant intensity $\lambda$ ;

(b)

with variable intensity $\lambda (t)$ .

Problem 33.6

Let $\{ \tau_{k}\}_{k=1}^{\infty }$ be the consecutive jump times of a Poisson process with intensity $\lambda$ . Prove that for any $T>0$

\sum _{k=1}^{\infty } \mathbb {P}\left(\tau _{k} \leq T\right) = \lambda T.

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Problem 33.7

(a)(b)(c)

Prove that the Poisson process:

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(a)

is differentiable in probability;

(b)

is differentiable in the sense of mean convergence of any order $p \in (0, 1)$ ;

(c)

is not differentiable in the sense of mean convergence of any order $p \geq 1$ .

Problem 33.8

Let there be a Poisson process with intensity $\lambda$ . Let the process $\xi (t)$ take the values $-1$ and $1$ , where over time $\xi (t)$ switches its value from $-1$ to $1$ and back at each jump of the Poisson process. Let $\mathbb {P}\left(\xi (0)=-1\right) = \mathbb {P}\left(\xi (0)=1\right) = 1/2$ . Find the covariance function of the process $\xi (t)$ . Is the process $\xi (t)$ stationary?

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Problem 33.9

Let there be a Poisson process with intensity $\lambda$ and a sequence of independent identically distributed random variables $\{ \eta_{n}\}$ , with the process and $\{ \eta_{n}\}$ independent of one another. Define the process $\xi (t)$ as follows: $\xi (t) = \eta_{n}$ if the time $t$ lies between the $n$ -th and $(n+1)$ -th jumps of the Poisson process. Find the covariance function of the process $\xi (t)$ . Is the process $\xi (t)$ stationary?

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Problem 33.10

(a)(b)

Points randomly placed in the plane form a Poisson field with intensity $\lambda$ , i.e., the number of points in any region $S$ of area $\operatorname {mes} S$ has a Poisson distribution with parameter $\lambda \operatorname {mes} S$ , and the numbers of points in disjoint regions are independent. Let $\rho_{1} \leq \rho_{2} \leq \ldots$ be the distances from the origin to the points of this field, arranged in increasing order.

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(a)

How can the sequence $\rho_{n}$ , $n=1,2,\ldots$ , be described?

(b)

Find the density of $\rho_{n}$ , the mean value of $\rho_{n}$ , and the asymptotics of this mean value as $n \to \infty$ .

Problem 33.11

Under the conditions of problem 33.10, prove that the coordinates $(\xi , \eta )$ of the point of the field closest to the origin are normally distributed, with means $m_{\xi } = m_{\eta } = 0$ and variances $\sigma_{\xi }^{2} = \sigma_{\eta }^{2} = 1/2\pi \lambda$ .

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Problem 33.12

Points randomly placed in three-dimensional space form a Poisson field with intensity $\lambda$ . Answer the questions of problem 33.10.

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Problem 33.13

Let $\pi (t)$ be a Poisson process with parameter 1. Find the distribution of the stochastic integral

\int _{0}^{t} s^{2}\, d(\pi (s) - s).

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Problem 33.14

(a)(b)(c)(d)

For a Poisson stream with intensity 7 events per unit time, find the probability that

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(a)

exactly 10 events occur in a unit of time;

(b)

at least two events occur in half a unit of time;

(c)

more than 10 units of time pass between the first and second events;

(d)

less than 1/4 of a unit of time passes between the second and third events.

Problem 33.15

(a)(b)

Let there be, on average, 4 accidents per month. Find the probability that

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(a)

over three months, at least one accident occurs every month;

(b)

at least three accidents occur over three months.

Problem 33.16

(a)(b)(c)

Let the arrival times of customers at a store form a Poisson process with intensity 1 per minute.

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(a)

What is the probability that no customers arrive during a five-minute interval of time?

(b)

Suppose each customer is a woman with probability 0.2 and a man with probability 0.8. What is the probability that more than two female customers arrive in a given three-minute interval, given that at least three male customers arrive?

(c)

Suppose every third customer receives a gift from the store. What is the distribution of the time between consecutive arrivals of two such customers? How can the probability that this time exceeds three minutes be found using properties of the Poisson process?

Problem 33.17

(a)(b)

Let telephone calls arrive in a Poisson stream with intensity $\lambda =4$ per minute. Find the probability that the 12th call arrives

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(a)

no later than two minutes;

(b)

between the 3rd and 5th minute.

Problem 33.18

(a)(b)

Let $\xi (t)$ be a Poisson process with parameter $\lambda$ . Let $0 < t_{1} < t_{2} < t_{3}$ and $0 < m < n$ . Find

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(a)

the distribution of the random variable $\xi (t_{2})$ given that $\xi (t_{3}) = n$ ;

(b)

the distribution of the random variable $\xi (t_{2}) - \xi (t_{1})$ given that $\xi (t_{1}) = m$ and $\xi (t_{3}) = n$ .

Problem 33.19

A train departs at time $s > 0$ . Starting from time $t=0$ , passengers arrive in a Poisson stream with intensity $\lambda$ . Prove that the mean total waiting time of the passengers equals $\lambda s^{2}/2$ .

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Problem 33.20

Suppose the times at which accidents occur that are the fault of a male driver and of a female driver form independent Poisson processes with intensities 10 and 2, respectively. Find the probability that the 8th accident caused by a male driver occurs before the 4th accident caused by a female driver.

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Problem 33.21

An insurance company has $n$ different insurance policies. Suppose the overall process of claim arrivals is Poisson with intensity $\lambda$ , and that each claim may pertain to each policy with the same probability $1/n$ . Find the expectation and variance of the number of distinct types of policies for which claims arrive during a unit of time.

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Problem 33.22

Vanya collects stamps. He very much wants to buy a certain set of two stamps issued in 1930, say a green one and a blue one. Suppose green stamps appear for sale (and then disappear) in a Poisson stream with intensity $\lambda_{1}$ (per year), while blue ones appear in an independent Poisson stream with intensity $\lambda_{2}$ . Find the expectation of the time needed to acquire at least one stamp of each color.

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