Random Processes

Let $\eta$ be a random variable with distribution function $F(x)$. Find all finite-dimensional distributions of the random process $\xi (t) = \eta + t$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let $\eta$ be a random variable uniformly distributed on the interval $[0, 1]$. Find all finite-dimensional distributions of the random process $\xi (t) = \; \mathbb {1}_{t < \eta }$, $t \in [0, 1]$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let the random variable $\xi$ have a normal distribution with expectation $a$ and variance $\sigma^{2}$; let $b \in \mathbb {R}$. Find all finite-dimensional distributions of the random process $\xi (t) = \xi t + b$, its expectation $m(t)$, and its covariance function $B(t, s)$.

Let $\phi$ be a random variable with density $\cos x$ for $x \in [0, \pi /2]$; let $a$ and $\omega$ be positive constants. Is the random process $\xi (t) = a\sin (\omega t + \phi )$ stationary?

Let $\xi$ and $\eta$ be uncorrelated random variables with zero means and finite variances, and let $\alpha , \beta > 0$ be constants. Find the expectation $m(t)$ and the covariance function $B(t, s)$ of the random process $\xi (t) = \xi e^{-\alpha t} + \eta e^{-\beta t}$.

Let $\xi$ and $\eta$ be uncorrelated random variables with zero means and unit variances. Find the expectation $m(t)$ and the covariance function $B(t, s)$ of the random process $\xi (t) = t + \xi \cos \gamma t + \eta \sin \gamma t$.

Let $\xi (t)$ be a differentiable random process with expectation $m(t)$ and covariance function $B(t, s)$. Find the expectation and the covariance function of its derivative $\dfrac {d\xi (t)}{dt}$.

Let a random process $\xi (t)$ have mean value $m(t) = t^{2} - 1$ and covariance function $B(t, s) = 2e^{-\alpha (s-t)^{2}}$. Find the expectation and the covariance function of the random process:

Let $\xi (t)$ be a random process with expectation $m(t)$ and covariance function $B(t, s)$; the sample paths of the process are continuous with probability 1. Find the expectation and the covariance function of the integral

Let a random process $\xi (t)$ equal $\eta \cos (\gamma t - \theta )$, where $\eta$ is a random variable with zero mean and variance $\sigma^{2}$; $\theta$ is a random variable uniformly distributed on the interval $[0, 2\pi ]$; $\gamma$ is a nonrandom parameter. The random variables $\eta$ and $\theta$ are independent. Find the expectation and the covariance function of the process $\xi (t)$. Under what conditions is $\xi (t)$ a stationary process?

Let a random process $\xi (t)$ equal $\sigma \cos 2\pi (\xi t + \eta )$, where $\sigma$ is a positive constant, $\xi$ is a random variable with density $f(x)$, where $0 \leq \xi \leq 1/2$, and $\eta$ is a random variable uniformly distributed on the interval $[-1/2, 1/2]$. The random variables $\xi$ and $\eta$ are independent. Prove that $\xi (t)$ is a stationary process. Find its covariance function and spectral density.

Let $\xi (t) = \eta_{1}f_{1}(t) + \ldots + \eta_{n}f_{n}(t)$, where $\eta_{1}, \ldots , \eta_{n}$ are uncorrelated random variables with zero means and variances $\sigma_{1}^{2}, \ldots , \sigma_{n}^{2}$ respectively, and $f_{1}, \ldots , f_{n}$ are nonrandom functions. Find the expectation and the covariance function of the process $\xi (t)$.

Show that there is no stationary process whose covariance function $b(s-t) = B(t, s)$ is constant and nonzero on some interval $(-t_{1}, t_{1})$ and equal to zero outside it.

Find the spectral density of a stationary process whose covariance function equals:

Let the spectral density of a stationary process equal

Find the covariance function of the process.

Let $\xi (t)$ be a stationary process, mean-square differentiable. Prove that for any fixed $t$, the values of the process $\xi (t)$ and of its derivative $\dfrac {d\xi (t)}{dt}$ are uncorrelated.

Let $\xi (t)$ and $\eta (t)$ be stationary processes defined on the same probability space. Is the sum $\xi (t) + \eta (t)$ a stationary process?

Let $\xi (t)$ and $\eta (t)$ be stationary processes defined on the same probability space, with $\{ \xi (t)\}$ independent of $\{ \eta (t)\}$. Is the sum $\xi (t) + \eta (t)$ a stationary process?

Let $\xi (t)$ be a stationary process. Determine the conditions under which the process $\xi (t)$ converges in probability as $t \to \infty$.

Let $\xi (t)$ be a random process such that all the $\xi (t)$ are independent of one another and have the same density $p$. Prove that this process is stochastically continuous at no point whatsoever.

Let $\xi$ and $\eta$ be random variables, where $\eta$ has a symmetric distribution and $\mathbb {P}\left(\eta = 0\right) = 0$. Find the probability that the realizations of the random process $\xi (t) = \xi + t(\eta + t)$, $t \geq 0$, are increasing.

Let $(\Omega , \mathscr {F}, \mathbb {P})$ be the probability space consisting of the interval $[0, 1]$ with the $\sigma$-algebra of Borel subsets and Lebesgue measure. Consider the random process $\xi (t, \omega )$ defined as follows:

Show that the process $\xi (t, \omega )$ is stochastically continuous, but all of its sample paths are discontinuous at every point.

Construct an example of a random process for which the set of elementary outcomes on which the process is continuous is not an event.

Let $w(t)$ be a Wiener process. Is the process $\eta (t)$ Markovian, if

Let $w(t)$ be a Wiener process. Is the process $\eta (t)$ Markovian, if

Let $w(t)$ be a Wiener process. Is the process $\eta (t) = w(t) - \left[w(t)\right]$ Markovian, where $\left[x\right]$ is the integer part of $x$?

Find the covariance function of the Wiener process.

Prove that the Wiener process is not differentiable in probability.

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = w(t) - tw(1)$, considered on the time interval $t \in [0, 1]$.

Let $w(t)$ be a standard Wiener process. Find the covariance function of the process $e^{-bt}w(ae^{2bt})$, where $a$ and $b$ are real numbers (this process is called the Ornstein--Uhlenbeck process).

Let $w(t)$ be a standard Wiener process. Find the joint distribution of $w(t)$ and $\displaystyle \int_{0}^{t} w(s)\, ds$, $t \geq 0$.

Let $w(t)$ be a standard Wiener process. Prove that the random process $w^{(1)}(t) = tw(1/t)$ is also a standard Wiener process.

Let $w(t)$ be a Wiener process with zero drift. Prove that the process

is also a Wiener process.

Let $w(t)$ be a three-dimensional Wiener process starting at zero. Prove that with probability 1

Let $w(t)$ be a standard Wiener process. Find the distributions of the following stochastic integrals:

Let $w(t)$ be a standard Wiener process. Determine which of the following processes are Wiener processes:

Let $w_{1}(t)$ and $w_{2}(t)$ be independent standard Wiener processes. For which $c_{1}$ and $c_{2}$ is the random process $c_{1}w_{1}(t) + c_{2}w_{2}(t)$ a standard Wiener process?

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = (1-t)w(t/(1-t))$, $t \in [0, 1)$, $w^{(0)}(1) = 0$.

Prove that the quadratic variation of the standard Wiener process $w(t)$ is finite, i.e., that

converges in mean square to $t$ as $n \to \infty$.

Let $w(t)$ be a standard Wiener process. Find the mean-square limit as $n \to \infty$ of:

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

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

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.

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 $t$).

Find the covariance function of a Poisson process:

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

Prove that the Poisson process:

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?

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?

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.

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 $\sigma_{\xi }^{2} = \sigma_{\eta }^{2} = 1/2\pi \lambda$.

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

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

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

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

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

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

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

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 $\lambda s^{2}/2$.

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.

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.

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.

Construct a stationary sequence whose covariance function $b(n)$ equals 1 for even $n$ and 0 for odd $n$.

Let $\xi$ be a random variable with the standard normal distribution. Set $\xi_{n} = \xi$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

Let $\xi$ and $\eta$ be independent random variables, each taking the values $-1$ and $1$ with probabilities $1/2$. Set $\xi_{2n+1} = \xi$ and $\xi_{2n} = \eta$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

Let $\xi$ and $\eta$ be uncorrelated random variables, each having the standard normal distribution. Set $\xi_{2n+1} = \xi$ and $\xi_{2n} = \eta$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be independent random variables uniformly distributed on the interval $[-2, 2]$. Set $\xi_{n} = \eta_{n-1}\eta_{n}$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$.

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be uncorrelated random variables with zero means and unit variances. Set $\xi_{n} = \eta_{n-3}+\eta_{n-2}+\eta_{n-1}+\eta_{n}$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{2}$ from the observations $\{ \xi (n), n \leq 0\}$.

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be independent random variables with the standard normal distribution. Set $\xi_{n} = \eta_{n-2}+\eta_{n-1}+\eta_{n}$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$.

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ and $\xi$ be independent random variables uniformly distributed on the interval $[-1, 1]$. Set $\xi_{n} = \eta_{n-1}+\eta_{n}+\xi$ for every integer $n$. Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$.

Let $w(u)$ be a standard Wiener process. Find the distributions of the following random sequences:

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, where $\xi_{0} = 1$ and $\mathbb {E}\left[\xi_{1}\right] = m$. Find $\mathbb {E}\left[\xi_{n+k} \mid \xi_{n}\right]$.

Find the generating function of the number of particles in the $n$-th generation, if the generating function of the offspring of a single particle equals $pz + 1 - p$.

Find the generating function of the number of particles in the second generation, if the generating function of the offspring of a single particle equals:

Find the extinction probability of a branching process, if the generating function of the number of offspring of a single particle equals:

In problem 35.4(a)--(c), find the distribution of the extinction time of the branching process.

Let $\xi_{n}$ be a branching process with initial population size $N$ and generating function $\phi (z) = 1-p+pz$. Find the distribution of the extinction time $\tau$ of the process.

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, $\xi_{0} = 1$. Prove that

Find the generating function of the total number of particles in the first $n$ generations, if the generating function of the immediate offspring of a single particle equals $pz + 1 - p$.

Let a particle have $k$ direct offspring with probability $p_{k} = bc^{k-1}$, $k=1,2,\ldots$, and $p_{0} = 1 - p_{1} - p_{2} - \ldots$, where $b, c > 0$ and $b + c < 1$. Find the generating function of the immediate offspring of a single particle.

Consider a branching process $\xi_{0}, \xi_{1}, \ldots$ with generating function

where $0 < c < b+c < 1$ and $1-b-c > c(1-c)$. Find

Let, in the branching process $\xi_{0}, \xi_{1}, \ldots$ of problem 35.10, the equality $1-b-c = c(1-c)$ hold. Find

Prove that the Poisson process with intensity $\lambda$ is a homogeneous jump Markov process. Find the transition function of this process.

Prove that the transition probabilities of a homogeneous jump Markov process satisfy the Kolmogorov--Chapman equation:

Prove that, in the differentiable case, the transition probabilities of a homogeneous jump Markov process with finitely many states satisfy the forward Kolmogorov equations

as well as the backward Kolmogorov equations

Let a homogeneous jump Markov process have two states 0 and 1, where, say, 0 means that some subject is healthy, and 1 that they are ill. Find the transition probabilities, if the transition intensity matrix equals