Random Variables

If $\left|\xi \right|$ is an $\mathcal{F}$-measurable function, is it true that $\xi$ is also $\mathcal{F}$-measurable?

Let $\xi , \eta$ be two random variables defined on $(\Omega , \mathcal{F})$. Let, moreover, $A \in \mathcal{F}$. Prove that the function

is also a random variable.

Let $\xi , \eta$ be two random variables defined on $(\Omega , \mathcal{F})$. We have

Prove that $\left\{ \xi =\eta \right\} \in \mathcal{F}$.

A random variable $\xi$ has an exponential distribution with parameter $\lambda$. Find the densities of the random variables:

1. $\xi^{k}, k \in \mathbb {N}$

2. $\frac{1}{\lambda } \ln \xi$

3. $\left\{ \xi \right\}$, where $\left\{ \cdot \right\}$ is the fractional part,

4. $1-e^{-\alpha \xi }$.

A random variable $\xi$ has a standard Cauchy distribution. Find the densities of the random variables

• $\frac{\xi^{2}}{1+\xi^{2}}$,

• $\frac{1}{1+\xi^{2}}$,

• $\frac{2 \xi }{1-\xi^{2}}$,

• $\frac{1}{\xi }$.

A random variable $\xi$ is uniformly distributed on the interval $(a, b)$, where $-\infty <a<b<\infty$. Find the density of the random variable $\xi^{2}$.

The density of the random vector $(\xi , \eta )$ equals

Find the densities of the random variables $\xi , \eta , \xi +\eta$.

A web graph $G$ is a directed graph whose vertices $1, \ldots , N$ correspond to pages on the Internet, and whose edges correspond to links. One of the most well-known search factors is called PageRank. The idea of PageRank is based on the following model of user behavior. At each moment of time $n \in \mathbb {N}$, the user either follows a link with probability 0.85 (with equal probability for each link), or chooses an arbitrary page with probability 0.15 (with equal probability for each page). At time $n=0$ the user chooses each page with probability $1 / N$. Let $\xi_{1}$ be the number of the page on which the user finds himself at time 1, and $\xi_{2}$ at time 2. If $G$ is a directed simple cycle or a complete graph with loops, devise a probability space and define on it the random vector $\left(\xi_{1}, \xi_{2}\right)$. Find its distribution.

Let $\xi$ be a random variable with continuous distribution function $F$. What is the distribution of the random variable $F(\xi )$?

Let $\xi_{1}, \xi_{2}, \ldots$ be an infinite Bernoulli scheme, where

for any $i \in \mathbb {N}$. Devise a probability space with random variables $\xi_{1}, \xi_{2}, \ldots$ defined on it and a function $f: \mathbb {R}^{\infty } \rightarrow \mathbb {R}$ such that