General Properties

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.