Math Problems Vault

Problem 3.1

Find the distribution function of the random variable $\xi$ defined on the probability space ( $\Omega , \mathscr {A}$ , Р), representing the interval $[0,1]$ with the $\sigma$ -algebra of Borel subsets and Lebesgue measure as the probability, if:

?

(a)

$\xi = \omega$ ;

(b)

$\xi = \omega^{2}$ ;

(c)

$\xi = \omega^{\alpha }, \alpha < 0$ ;

(d)

$\xi = \sin \pi \omega$ ;

(e)

$\xi = \begin{cases} 2 \omega , & 0 \leqslant \omega \leqslant 1 / 2, \\ 2(1-\omega ), & 1 / 2 < \omega \leqslant 1 ; \end{cases}$

(f)

$\xi = \begin{cases} \omega , & 0 \leqslant \omega \leqslant 1 / 3, \\ -1, & 1 / 3 < \omega \leqslant 2 / 3, \\ -\omega^{3}, & 2 / 3 < \omega \leqslant 1 ; \end{cases}$

(g)

$\xi = \begin{cases} 1 / 4, & 0 \leqslant \omega \leqslant 1 / 4, \\ 1, & 1 / 4 < \omega \leqslant 3 / 4, \\ 1 / 4, & 3 / 4 < \omega \leqslant 1. \end{cases}$

Problem 3.2

(a)(b)(c)

Let $\xi$ and $\eta$ be random variables defined on the probability space ( $\Omega , \mathscr {A}, \mathbb {P}$ ), where $\Omega$ is the square with vertices $(0,0),(0,1),(1,0),(1,1)$ , $\mathscr {A}$ is the $\sigma$ -algebra of Borel subsets of $\Omega$ , $\mathbb {P}$ is Lebesgue measure. Find the distribution function and the density of the distribution of the random variable $\xi +\eta$ , if:

?

(a)

$\xi = \omega_{1}+\omega_{2}, \eta = \omega_{1}-\omega_{2} ;$

(b)

$\xi = \omega_{1}, \eta = \omega_{2} ;$

(c)

$\xi = 1$ at $\omega_{1} = \omega_{2}, \xi = 0$ at $\omega_{1} \neq \omega_{2}, \eta = \omega_{1} \omega_{2}$ .

Are the random variables $\xi$ and $\eta$ independent?

Problem 3.3

(a)(b)

Let the random variable $\xi$ be defined on the probability space ( $\Omega , \mathscr {A}, P$ ), where $\Omega$ is the triangle with vertices at the points $(0,0),(2,1),(2,0)$ , $\mathscr {A}$ is the $\sigma$ -algebra of Borel subsets of the indicated triangle, $\mathbb {P}$ is Lebesgue measure. Find the distribution function and the density of the distribution of the random variable $\xi$ , if:

?

(a)

$\xi = \omega_{1}$ ;

(b)

$\xi = \omega_{2}$ .

Problem 3.4

(a)(b)

An isosceles triangle is formed by a unit vector in the direction of the abscissa axis and a unit vector in a random direction. Find the distribution function of the length of the third side:

?

(a)

in $\mathbb {R}^{2}$ ;

(b)

in $\mathbb {R}^{3}$ .

Problem 3.5

A circle of unit radius centered at the origin has its north pole on the positive abscissa semi-axis. From the pole a ray is directed at random, with its angle to the abscissa axis distributed uniformly on the interval $\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ . Find the distribution function of the length of the chord inside the circle.

?

Problem 3.6

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

Let $\xi$ and $\eta$ be independent random variables with distribution functions $F(x)$ and $G(x)$ respectively. Find the distribution functions of the following random variables:

?

(a)

$\max \left\{ \xi , \eta \right\}$ ;

(b)

$\min \left\{ \xi , \eta \right\}$ ;

(c)

$\max \left\{ 2 \xi , \eta \right\} ;$

(d)

$\min \left\{ \xi^{3}, \eta \right\}$ .

Problem 3.7

Let $\xi$ and $\eta$ be independent random variables having the same exponential distribution with parameter $\alpha$ . Find the distribution functions and densities of distribution of the following random variables:

?

(a)

$\xi^{3}$ ;

(b)

$\xi -\eta$ ;

(c)

$\max \left\{ \xi , \eta^{3}\right\}$ ;

(d)

$\min \left\{ \xi , \eta^{3}\right\}$ ,

(e)

$3+2 \xi$ ;

(f)

$\left|\xi -\eta \right|$ .

Problem 3.8

Solve the previous problem under the assumption that $\xi$ and $\eta$ are uniformly distributed on the interval $[-1,1]$ .

?

Problem 3.9

Let $\xi$ be a random variable having an exponential distribution with parameter $\lambda$ , and let $f(x)$ be a positive, strictly monotone, differentiable function. Find the density of the distribution of the random variable $f(\xi )$ .

?

Problem 3.10

(a)(b)(c)

Let $\xi , \eta$ and $\zeta$ be independent random variables, where $\zeta$ takes the values 1 and 0 with probabilities $p$ and $q$ respectively, $p+q = 1$ , and $\xi$ and $\eta$ have distribution functions $F(x)$ and $G(x)$ . Find the distribution functions of the following random variables:

?

(a)

$\zeta \xi +(1-\zeta ) \eta$ ;

(b)

$\zeta \xi +(1-\zeta )(\max \left\{ \xi , \eta \right\} )$ ;

(c)

$\zeta \xi +(1-\zeta )(\min \left\{ \xi , \eta \right\} )$ .

Problem 3.11

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables, $\mathbb {P}\left(\xi_{i} = 1\right) = \mathbb {P}\left(\xi_{i} = -1\right) = 1 / 2, i = 1,2, \ldots , n$ . Find the distribution of the random variable

\eta _{n} = \prod _{i = 1}^{n} \xi _{i} .

?

Problem 3.12

(a)(b)(c)

Can a constant $c$ be chosen so that the function $c x^{-3}$ defines a probability density on:

?

(a)

the ray $[1,+\infty )$ ;

(b)

the ray $[0,+\infty )$ ;

(c)

the interval $[-2,-1]$ .

Problem 3.13

Let $\xi$ and $\eta$ be independent identically distributed random variables, and let $\varphi (x, y)$ be a Borel function of two variables. Prove that the random variables $\varphi (\xi , \eta )$ and $\varphi (\eta , \xi )$ are identically distributed. Can the independence assumption be dropped?

?

Problem 3.14

Let $\xi$ be a random variable with a symmetric distribution, and let $A$ be a Borel set on the line symmetric about zero. Set

\eta = \begin{cases} \xi , & \text{ if } \xi \in A, \\ -\xi , & \text{ if } \xi \notin A . \end{cases}

Prove that the random variables $\xi$ and $\eta$ are identically distributed.

?

Problem 3.15

Let $\xi$ and $\eta$ be random variables with distribution functions $F(x)$ and $G(x)$ respectively. Prove that if $\mathbb {P}\left(\xi > \eta \right) = 1$ , then $F(x) < G(x)$ for all $x$ . Is the converse true, given that $\xi$ and $\eta$ are defined on the same probability space?

?

Problem 3.16

(Continuation) Let $F(x)$ and $G(x)$ be two distribution functions, with $F(x) < G(x)$ for all $x$ . Prove that there exists a probability space and two random variables $\xi$ and $\eta$ , defined on this probability space, such that $F(x) = \mathbb {P}\left(\xi \leqslant x\right), G(x) = \mathbb {P}\left(\eta \leqslant x\right)$ and $\xi > \eta$ .

?

Problem 3.17

Prove that the set of discontinuity points of any distribution function is at most countable.

?

Problem 3.18

Can the set of discontinuity points of a distribution function be everywhere dense on the line?

?

Problem 3.19

Prove that any distribution function $F(x)$ can be represented in the form

F(x) = a_{1} F_{1}(x)+a_{2} F_{2}(x), \quad a_{1} \geqslant 0, \quad a_{2} \geqslant 0, \quad a_{1}+a_{2} = 1,

where $F_{1}(x)$ and $F_{2}(x)$ are distribution functions, with $F_{1}(x)$ continuous and $F_{2}(x)$ a discrete distribution function. Prove that such a representation is unique.

?

Problem 3.20

Let the Borel function of two variables $F(x, y)$ , for each fixed $y$ , be a distribution function. Prove that if $G(x)$ is a distribution function, then the function

H(x) = \int _{-\infty }^{\infty } F(x, y) d G(y)

is also a distribution function.

?

Problem 3.21

The distribution function $F(x)$ of the random variable $\xi$ is continuous at zero. Find the distribution of the random variable

\eta = \begin{cases} \frac{\xi }{\left|\xi \right|}, & \text{ if } \quad \xi \neq 0, \\ 1, & \text{ if } \quad \xi = 0. \end{cases}

?

Problem 3.22

The random variable $\xi$ has distribution function $F(x)$ . Find the distribution function of the random variable $\frac{1}{2}(\xi +\left|\xi \right|)$ .

?

Problem 3.23

Prove that if a distribution function is continuous at every point of the line, then it is uniformly continuous on the whole line.

?

Problem 3.24

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables taking the values 0 and 1 with probability $1 / 2$ each. Find the distribution of the random variable

\eta = \sum _{k = 1}^{\infty } \frac{\xi _{k}}{2^{k}} .

?

Problem 3.25

Let $\xi$ be a random variable having the uniform distribution on the interval $[0,1]$ . Prove that for any distribution function $F(x)$ there exists a Borel function $f(x)$ such that $F(x)$ is the distribution function of the random variable $f(\xi )$ .

?

Problem 3.26

Let $\xi$ be a random variable uniformly distributed on the interval $[0,1)$ , and let $\xi = \frac{\delta_{1}}{2}+\frac{\delta_{2}}{2^{2}}+\frac{\delta_{3}}{2^{3}}+\ldots , \delta_{n} = 0$ or 1, be the binary expansion of $\xi$ . Prove that for any natural number $n$

\mathbb {P}\left(\delta _{n} = 0\right) = \mathbb {P}\left(\delta _{n} = 1\right) = \frac{1}{2}

and the random variables $\delta_{1}, \delta_{2}, \ldots$ are mutually independent.

?

Problem 3.27

Let the conditions of the previous problem hold. Let us arrange the digits of the binary expansion of $\xi$ — the random variables $\delta_{1}, \delta_{2}, \ldots$ — in the form of the following square table:

\begin{aligned} & \delta _{1} \delta _{3} \delta _{6} \ldots \\ & \delta _{2} \delta _{5} \ldots \\ & \delta _{4} \ldots \\ & \delta _{7} \ldots \end{aligned}

From the digits of the $k$ -th row of this table let us construct a binary expansion defining a random variable $\xi_{k}$ , for example:

\begin{aligned} & \xi _{1} = \frac{\delta _{1}}{2}+\frac{\delta _{3}}{2^{2}}+\ldots \\ & \xi _{2} = \frac{\delta _{2}}{2}+\frac{\delta _{5}}{2^{2}}+\ldots \\ & \xi _{3} = \frac{\delta _{4}}{2}+\frac{\delta _{8}}{2^{2}}+\ldots \end{aligned}

Prove that the random variables $\xi_{1}, \xi_{2}, \ldots$ are independent and each of them has the uniform distribution on the interval $[0,1]$ .

?

Problem 3.28

Let ( $\Omega , \mathscr {A}, P$ ) be the probability space representing the interval $[0,1]$ with the $\sigma$ -algebra of Borel subsets and Lebesgue measure, and let $F_{1}(x), F_{2}(x), \ldots$ be an arbitrary sequence of distribution functions. Prove that on the indicated probability space there exists a sequence of independent random variables $\xi_{1}, \xi_{2}, \ldots$ such that for each $n$ the distribution function of the random variable $\xi_{n}$ coincides with $F_{n}(x)$ .

?

Problem 3.29

Let $F(x)$ and $G(x)$ be distribution functions. Find necessary and sufficient conditions for the function $H(x) = F(G(x))$ to be a distribution function.

?

Problem 3.30

Give an example of a random variable $\xi$ with density $p(x)$ and a continuous function $g(x)$ such that $g(\xi )$ is a nondegenerate random variable with a discrete distribution.

?

Problem 3.31

Let $\mathbb {P}$ be an arbitrary probability distribution on the line. Prove that for any Borel set $A$ and any positive $\varepsilon$ there exist an open set $G$ and a closed set $F$ such that $F \subseteq A \subseteq G$ and $\mathbb {P}\left(G \backslash F\right) < \varepsilon$ .

?

Problem 3.32

Let $\mathbb {P}$ be an arbitrary probability distribution on the line. Prove that for any Borel set $A$ and any positive $\varepsilon$ there exists a compact set $K$ such that $K \subseteq A$ and $\mathbb {P}\left(A \backslash K\right) < \varepsilon$ .

?

Problem 3.33

What is the cardinality of the set of all distribution functions?

?

Problem 3.34

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with distribution functions $F_{1}(x), F_{2}(x), \ldots$ respectively, and let $v$ be a positive integer-valued random variable independent of all $\xi_{1}, \xi_{2}, \ldots$ . Set $p_{k} = \mathbb {P}\left(v = k\right), k = 1,2, \ldots$ . Find the distribution function of the random variable $\xi$ .

?

Problem 3.35

Prove that for any continuous distribution function $F(x)$

\int _{-\infty }^{\infty } F(x) d F(x) = \frac{1}{2}

?

Problem 3.36

Prove that for any continuous distribution function $F(x)$ and any natural numbers $n$ and $k$

\int _{-\infty }^{\infty } F^{k}(x) d F^{n}(x) = \frac{n}{(n+k)} .

?

Problem 3.37

Let $\xi$ be a random variable with continuous distribution function $F(x)$ . Find the distribution function of the random variable $F(\xi )$ .

?

Problem 3.38

Prove that if the random variable $\xi$ has an absolutely continuous distribution, then the random variable $\left|\xi \right|$ also has an absolutely continuous distribution. Is the converse assertion true?

?

Problem 3.39

The random variables $\xi$ and $\eta$ are independent and have the same distribution:

\mathbb {P}\left(\xi = k\right) = \mathbb {P}\left(\eta = k\right) = 1 / N, \quad k = 1, \ldots , N .

Set

v = \min \left\{ \xi , \eta \right\} , \quad \mu = \max \left\{ \xi , \eta \right\} , \quad \lambda = \mu -v .

Find the distributions of the random variables $\nu , \mu$ and $\lambda$ .

?

Problem 3.40

Let $\xi$ and $\eta$ be independent random variables with continuous distribution functions $F(x)$ and $G(x)$ respectively. Find the distribution function of the product $\xi \eta$ .

?

Problem 3.41

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables, and let $F(x)$ be the distribution function of $\xi_{1}$ . Let us introduce a function of two variables $\varphi (x, y)$ as follows:

\varphi (x, y) = \begin{cases} 1, & \text{ at } x \geqslant y, \\ 0, & \text{ at } x < y . \end{cases}

For each fixed $x$ , find the distribution of the random variable

\eta _{n} = \frac{1}{n} \sum _{k = 1}^{n} \varphi \left(x, \xi _{k}\right)

?

Problem 3.42

The distribution function $F(x)$ of a nonnegative random variable is called subadditive if

F(x+y) \leqslant F(x)+F(y)

for any $x, y \geqslant 0$ . Give an example of a subadditive distribution function.

?

Problem 3.43

Let $p(x)$ be a distribution density, nonincreasing for $x \geqslant 0$ and equal to zero for $x < 0$ . Prove that the corresponding distribution function is subadditive (for the definition see the previous problem).

?

Problem 3.44

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

Prove that for any distribution function $F(x)$ the following relations hold:

?

(a)

$\lim_{x \rightarrow +\infty } x \int_{x}^{\infty } \frac{d F(y)}{y} = 0$ ,

(b)

$\lim_{x \rightarrow -\infty } x \int_{-\infty }^{x} \frac{d F(y)}{y} = 0$ ,

(c)

$\lim_{x \rightarrow -0} x \int_{-\infty }^{x} \frac{d F(y)}{y} = 0$ ,

(d)

$\lim_{x \rightarrow +0} x \int_{x}^{\infty } \frac{d F(y)}{y} = 0$ .

Problem 3.45

Let the distribution function $F(x)$ be continuous at zero. Prove that the function

p(x) = \begin{cases} \int _{x}^{+\infty } \frac{d F(y)}{y}, & \text{ at } x > 0 \\ -\int _{-\infty }^{x} \frac{d F(y)}{y}, & \text{ at } x < 0 \end{cases}

is a distribution density.

?

Problem 3.46

Prove that if in the previous problem we drop the condition of continuity at zero, i.e., assume that $0 < F(0)$ - $-F(0-0) = \alpha < 1$ , then the function $p(x) /(1-\alpha )$ is a probability distribution density.

?

Problem 3.47

Let $F_{1}(x)$ and $F_{2}(x)$ be two distribution functions. Prove that $F_{1}(x) \leqslant F_{2}(x)$ for all $x$ if and only if

\int _{-\infty }^{\infty } f(x) d F_{1}(x) \geqslant \int _{-\infty }^{\infty } f(x) d F_{2}(x)

for any monotonically nondecreasing function $f(x)$ for which the indicated integrals exist.

?

Problem 3.48

Let the real function $f(x)$ be such that for any two distribution functions $F_{1}(x)$ and $F_{2}(x)$ satisfying the inequality $F_{1}(x) \leqslant F_{2}(x)$ , the inequality

\int _{-\infty }^{\infty } f(x) d F_{1}(x) \geqslant \int _{-\infty }^{\infty } f(x) d F_{2}(x)

holds.

Prove that $f(x)$ is monotonically nondecreasing.

?

Problem 3.49

A distribution $P$ is called dominating for a family of distributions $\mathfrak {B}$ if for any Borel set $A$ such that $\mathbb {P}\left(A\right) = 0$ , the equalities $Q(A) = 0$ hold for all $Q$ from $\mathfrak {B}$ . A family of distributions is called dominated if there exists a distribution dominating it.

Prove that any countable family of distributions is dominated.

?

Problem 3.50

Give an example of a non-dominated family of distributions.

?

Problem 3.51

Prove that if $\mathfrak {H}_{1}, \mathfrak {B}_{2}, \ldots$ is a sequence of dominated families of distributions, then their union

\bigcup _{n = 1}^{\infty } \mathfrak {Z}_{n} .

is also dominated.

?

Problem 3.52

The convex hull of a family of distributions $\mathfrak {B}$ is the set of all distributions of the form

\sum _{i = 1}^{\infty } \alpha _{i} P_{i}

where $\alpha_{i} \geqslant 0, \sum_{i = 1}^{\infty } \alpha_{i} = 1, P_{i} \in \mathfrak {B}, i = 1,2, \ldots$ . Prove that the convex hull of a dominated family of distributions is dominated.

?

Problem 3.53

Prove that a family of distributions $\mathfrak {B}$ is dominated if and only if there exists an at most countable subfamily $\mathfrak {H}^{\prime }$ of the family $\mathfrak {H}$ such that if a Borel set $A$ satisfies the condition $\mathbb {P}\left(A\right) = 0$ for every $P \in \mathfrak {H}^{\prime }$ , then $Q(A) = 0$ for every $Q \in \mathfrak {B}$ .

?

Problem 3.54

The concentration function of a random variable $\xi$ is the function

Q_{\xi }(x) = \sup _{a} \mathbb {P}\left(a \leqslant \xi \leqslant a+x\right)

Clearly, the concentration function is uniquely determined by the distribution function. Is the converse true?

?

Problem 3.55

Prove that for any random variable $\xi$ and any positive $x$ there exists $a$ such that

Q_{\xi }(x) = \mathbb {P}\left(a \leqslant \xi \leqslant a+x\right) .

?

Problem 3.56

Prove that for any random variable $\xi$ and any nonnegative $\alpha$ and $x$ the inequality

Q_{\xi }(\alpha x) \leqslant ([\alpha ]+1) Q_{\xi }(x),

holds, where $[\alpha ]$ is the integer part of the number $\alpha$ .

?

Problem 3.57

(a)(b)

Let $F(x)$ be a distribution function, $Q(x)$ the corresponding concentration function. 1. Can $F(x)$ have more discontinuity points than $Q(x)$ ? 2. Can $Q(x)$ have more discontinuity points than $F(x)$ ?

?

(a)

(b)

Problem 3.58

Can a distribution function have an infinite number of discontinuity points while the corresponding concentration function has a finite number?

?

Problem 3.59

Let $F(x)$ be a distribution function, $Q(x)$ the corresponding concentration function, $n$ the number of discontinuity points of $F(x)$ , and $m$ the number of discontinuity points of $Q(x)$ . Prove that

m \leqslant C_{n}^{2}+1.

?

Problem 3.60

Let $\xi$ be an arbitrary random variable. Prove that for any positive $a$ and $l$ the set

A_{l, a} = \left\{ x: \mathbb {P}\left(x \leqslant \xi \leqslant x+l\right) \geqslant a\right\}

is compact (possibly empty).

?

Problem 3.61

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

Find the mathematical expectation and variance of the random variable $\xi = \xi (\omega )$ defined on the probability space ( $\Omega , \mathscr {A}, P$ ), representing the interval $[0,1]$ with the $\sigma$ -algebra of Borel subsets and Lebesgue measure, if:

?

(a)

$\xi = \omega^{2}$ ,

(b)

$\xi = \omega -1 / 2$ ,

(c)

$\xi = \sin \pi \omega$ ,

(d)

$\xi = \sin 2 \pi \omega$ .

Problem 3.62

The diameter $d$ of a circle is measured approximately, and it is known only that $0 < a \leqslant d \leqslant b$ . Regarding $d$ as a random variable uniformly distributed on the interval $[a, b]$ , find the mathematical expectation and variance of the area of the circle.

?

Problem 3.63

Two dice are thrown. Find the mathematical expectation of the sum of the points shown, given that it is known that different faces came up.

?

Problem 3.64

Prove that the variance $\operatorname {Var}\left[\xi \right] = 0$ vanishing is equivalent to the random variable $\xi$ being equal, with probability one, to a constant: $\mathbb {P}\left(\xi = c\right) = 1$ for some number $c$ .

?

Problem 3.65

Let $\xi$ and $\eta$ be independent identically distributed random variables with finite variance. Prove that $\xi$ takes values of one sign with probability one if and only if the random variables $\xi -\eta$ and $\left|\xi \right|-\left|\eta \right|$ are identically distributed.

?

Problem 3.66

Let $\xi$ be a random variable such that $\mathbb {P}\left(0 < \xi < 1\right) = 1$ . Prove that $\operatorname {Var}\left[\xi \right] < \mathbb {E}\left[\xi \right]$ .

?

Problem 3.67

(a)(b)

Let the random variables $\xi$ and $\eta$ be independent and $\mathbb {E}\left[\xi \right] = 1$ , $\mathbb {E}\left[\eta \right] = 2, \operatorname {Var}\left[\xi \right] = 1, \operatorname {Var}\left[\eta \right] = 4$ .

Find the mathematical expectations of the random variables:

?

(a)

$\xi^{2}+2 \eta^{2}-\xi \eta -4 \xi +\eta +4$ ;

(b)

$(\xi +\eta +1)^{2}$ .

Problem 3.68

Prove that for any random variables $\xi$ and $\eta$ having finite variances, the inequalities

(\sqrt{\operatorname {Var}\left[\xi \right]}-\sqrt{\operatorname {Var}\left[\eta \right]})^{2} \leqslant \operatorname {Var}\left[\xi +\eta \right] \leqslant (\sqrt{\operatorname {Var}\left[\xi \right]}+\sqrt{\operatorname {Var}\left[\eta \right]})^{2} .

hold.

?

Problem 3.69

Prove that if $\mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty , \alpha > 0$ , then $\mathbb {E}\left[\left|\xi \right|^{\beta }\right] < \infty$ for $0 < \beta \leqslant \alpha$ .

?

Problem 3.70

Give an example of a distribution having no moments of either positive or negative order.

?

Problem 3.71

Let $\xi$ and $\eta$ be identically distributed random variables. Is it true that

\mathbb {E}\left[\frac{\xi }{\xi +\eta }\right] = \mathbb {E}\left[\frac{\eta }{\xi +\eta }\right] ?

?

Problem 3.72

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed positive random variables. Prove that

\mathbb {E}\left[\frac{\xi _{1}+\ldots +\xi _{k}}{\xi _{1}+\ldots +\xi _{n}}\right] = \frac{k}{n}

for any $1 \leqslant k \leqslant n$ . Can the independence assumption be dropped?

?

Problem 3.73

(a)(b)

A random variable $\xi$ takes the values $0,1, \ldots , n, \ldots$ with probabilities decreasing in geometric progression:

?

(a)

find the relation between $\mathbb {E}\left[\xi \right]$ and $\operatorname {Var}\left[\xi \right]$ ;

(b)

it is known that $\mathbb {E}\left[\xi \right] = a$ ; find $\mathbb {P}\left(\xi = n\right), n = 0,1_{1} \ldots$ .

Problem 3.74

Let the random variable $\xi$ take a finite number of nonnegative values $x_{1}, \ldots , x_{r}$ . Prove that

\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathbb {E}\left[\xi ^{n+1}\right]}{\mathbb {E}\left[\xi ^{n}\right]} & = \max \left(x_{1}, \ldots , x_{r}\right) \\ \lim _{n \rightarrow \infty } \sqrt[n]{\mathbb {E}\left[\xi ^{n}\right]} & = \max \left(x_{1}, \ldots , x_{r}\right) \end{aligned}

?

Problem 3.75

Let $\xi$ be a nonnegative integer-valued random variable with finite mathematical expectation. Prove that

\mathbb {E}\left[\xi \right] = \sum _{i = 1}^{\infty } \mathbb {P}\left(\xi \geqslant i\right) .

?

Problem 3.76

$n$ letters addressed to different recipients are written. There are $n$ envelopes with the corresponding addresses. The letters are placed into the envelopes in random order. Let $\xi_{n}$ be the number of letters sent to the recipients they were intended for. Find $\mathbb {E}\left[\xi_{n}\right]$ .

?

Problem 3.77

Let $\xi$ be a random variable bounded with probability one: $\mathbb {P}\left(\left|\xi \right| \leqslant c\right) = 1$ . Prove that

\operatorname {Var}\left[\xi \right] \leqslant c \mathbb {E}\left[\left|\xi \right|\right] .

?

Problem 3.78

Prove that $\mathbb {E}\left[\xi \right]$ exists if and only if $\mathbb {E}\left[\left[\xi \right]\right]$ exists ( $[\cdot ]$ the integer part), and moreover $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\left[\xi \right]\right]$ if and only if $\xi$ is an integer-valued random variable.

?

Problem 3.79

Let $\mathbb {E}\left[\xi \right] = 0$ and $\mathbb {E}\left[\left|\xi \right|\right] = 1$ . Find $\mathbb {E}\left[\max \left\{ 0, \xi \right\} \right]$ and $\mathbb {E}\left[\min \left\{ 0, \xi \right\} \right]$ .

?

Problem 3.80

What conditions must the numbers $a$ and $b$ satisfy in order for there to exist a random variable $\xi$ such that $\mathbb {E}\left[\xi \right] = a$ , $\mathbb {E}\left[\left|\xi \right|\right] = b$ ?

?

Problem 3.81

Prove Jensen's inequality.

?

Problem 3.82

Let $\xi$ and $\eta$ be independent random variables with finite variances. Prove that

\operatorname {Var}\left[\xi \eta \right] \geqslant \operatorname {Var}\left[\xi \right] \cdot \operatorname {Var}\left[\eta \right] .

?

Problem 3.83

What conditions must the independent random variables $\xi$ and $\eta$ satisfy in order for the equality

\operatorname {Var}\left[\xi \eta \right] = \operatorname {Var}\left[\xi \right] \cdot \operatorname {Var}\left[\eta \right] ?

to hold?

?

Problem 3.84

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables, $\mathbb {E}\left[\xi_{i}\right] = m_{i}(i = 1,2, \ldots )$ ; let $A_{1}, A_{2}, \ldots$ be pairwise incompatible events, $\bigcup_{j = 1}^{\infty } A_{j} = \Omega , \mathbb {P}\left(A_{j}\right) = p_{j}$ ; let the random variables $\xi_{i}$ and $I_{A_{j}}$ be independent; let the random variable $\xi$ equal $\sum_{i = 1}^{\infty } \xi_{i} I_{A_{i}}$ ; let the random variable $\mu$ take the values $m_{i}$ with probabilities $p_{i}$ , respectively. Prove that

\operatorname {Var}\left[\xi \right] = \sum _{i = 1}^{\infty } p_{i} \operatorname {Var}\left[\xi _{i}\right]+\operatorname {Var}\left[\mu \right] .

?

Problem 3.85

The random variable $\xi$ has a finite absolute moment of order $p > 0$ :

\mathbb {E}\left[\left|\xi \right|^{p}\right] < \infty .

Prove that

t^{p} \mathbb {P}\left(\left|\xi \right| > t\right) \rightarrow 0

as $t \rightarrow \infty$ .

?

Problem 3.86

Prove that the distribution function $F(x)$ has a finite absolute moment of order $\alpha > 0$ if and only if the function $\left|x\right|^{\alpha -1}(1-F(x)+F(-x))$ is integrable on the whole real line.

?

Problem 3.87

Prove that for the random variable $\xi$ to have a finite absolute moment of order $\alpha > 0$ : $\mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty$ , it is necessary and sufficient that the series

\sum _{n = 1}^{\infty } n^{\alpha -1} \mathbb {P}\left(\left|\xi \right| \geqslant n\right) .

converge.

?

Problem 3.88

Let $\xi$ and $\eta$ be independent random variables taking nonnegative integer values, with $E\left|\xi \right| < \infty$ . Prove that

\mathbb {E}\left[\min \left\{ \xi , \eta \right\} \right] = \sum _{i = 1}^{\infty } \mathbb {P}\left(\xi \geqslant i\right) \mathbb {P}\left(\eta \geqslant i\right) .

?

Problem 3.89

Let $\xi$ be a nonnegative random variable with distribution function $F(x)$ and finite mathematical expectation.

Prove that

\mathbb {E}\left[\xi \right] = \int _{0}^{\infty }(1-F(x)) d x .

?

Problem 3.90

Let $\xi$ be a positive random variable with distribution function $F(x)$ and

\mathbb {E}\left[\frac{1}{\xi ^{\alpha }}\right] < \infty , \quad \alpha > 0.

Prove that

\frac{F(x)}{x^{\alpha }} \rightarrow 0

as $x \rightarrow 0$ .

?

Problem 3.91

Let $\xi$ be a nonnegative random variable with distribution function $F(x)$ and a finite moment of order $\alpha > 0$ . Prove that

\mathbb {E}\left[\xi ^{\alpha }\right] = \alpha \int _{0}^{\infty } x^{\alpha -1}(1-F(x)) d x .

?

Problem 3.92

Let $\xi$ be a positive random variable with distribution function $F(x)$ and a finite moment of order $\alpha < 0$ . Prove that

\mathbb {E}\left[\xi ^{\alpha }\right] = \left|\alpha \right| \int _{0}^{\infty } x^{\alpha -1} F(x) d x .

?

Problem 3.93

Prove that if $F(x)$ is a distribution function with finite mathematical expectation and such that $F(0) = 0$ , then the function

G(x) = \prod _{n = 1}^{\infty } F(x+n)

is a distribution function.

?

Problem 3.94

Let $\xi$ be a random variable with zero mathematical expectation and finite variance. Prove that

\mathbb {E}\left[\left|\xi \right|\right] \leqslant \frac{1}{2}(\operatorname {Var}\left[\xi \right]+1) .

?

Problem 3.95

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variance. Set

a_{n} = \mathbb {E}\left[\left|\frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{n}}\right|\right], \quad n = 1,2, \ldots

Prove that the sequence $a_{1}, a_{2}, \ldots$ is uniformly bounded.

?

Problem 3.96

The random variables $\xi_{1}, \xi_{2}, \ldots , \xi_{n}$ are independent and identically distributed. Each of the random variables $\xi$ takes two values, 1 and 0, with probabilities $p$ and $q = 1-p$ respectively. Let $\eta_{h}$ be the random variable equal to 0 if $\xi_{h} = \xi_{h+1}$ , and 1 if $\xi_{h} \neq \xi_{h+1}$ . Set $\zeta_{n} = \eta_{1}+\ldots +\eta_{n-1}$ . Find the mathematical expectation and variance of the random variable $\zeta_{n}$ .

?

Problem 3.97

The random variable $\xi$ is uniformly distributed on the interval $[a, b]$ . Find $a$ and $b$ , if $\mathbb {E}\left[\xi^{2}\right] = 1$ and $\mathbb {E}\left[\xi \right] = -\mathbb {E}\left[\xi^{3}\right]$ .

?

Problem 3.98

Let $\xi$ be a random variable with a distribution symmetric about zero. Prove that for any real $a$

\mathbb {E}\left[\left|\xi +a\right|\right] \geqslant \mathbb {E}\left[\left|\xi \right|\right] .

?

Problem 3.99

Show that there exist two random variables $\xi$ and $\eta$ such that $\mathbb {E}\left[\left|\xi \right|\right] < \infty , \mathbb {E}\left[\left|\eta \right|\right] < \infty$ and

\mathbb {E}\left[\left|\xi +a\right|\right] > \mathbb {E}\left[\left|\eta +a\right|\right]

for every real $a$ .

?

Problem 3.100

Let $\xi$ and $\eta$ be random variables such that for every $a$

\mathbb {E}\left[\left|\xi +a\right|^{\alpha }\right] = \mathbb {E}\left[\left|\eta +a\right|^{\alpha }\right]

( $\alpha$ some real number), and let $\zeta$ be a random variable independent of $\xi$ and $\eta$ . Prove that

\mathbb {E}\left[\left|\xi +\zeta \right|^{\alpha }\right] = \mathbb {E}\left[\left|\eta +\zeta \right|^{\alpha }\right]

?

Problem 3.101

Let $\xi$ and $\eta$ be random variables such that for every $a$

\mathbb {E}\left[\left|\xi +a\right|^{\alpha }\right] \leqslant \mathbb {E}\left[\left|\eta +a\right|^{\alpha }\right]

( $\alpha$ some real number), and let $\zeta$ be a random variable independent of $\xi$ and $\eta$ . Prove that

\mathbb {E}\left[\left|\xi +\zeta \right|^{\alpha }\right] \leqslant \mathbb {E}\left[\left|\eta +\zeta \right|^{\alpha }\right]

?

Problem 3.102

(a)(b)

Let $\xi$ and $\eta$ be nonnegative random variables with distribution functions $F(x)$ and $G(x)$ respectively, with $F(x) \geqslant G(x)$ for all $x$ . Prove that:

?

(a)

$\mathbb {E}\left[\xi^{\alpha }\right] \leqslant \mathbb {E}\left[\eta^{\alpha }\right], 0 \leqslant \alpha < \infty$ ;

(b)

$\mathbb {E}\left[\xi^{\alpha }\right] \geqslant \mathbb {E}\left[\eta^{\alpha }\right], \alpha < 0$ , if the indicated moments exist.

Problem 3.103

Let $\xi$ be a random variable with finite mathematical expectation. Prove that for any $x$

\max \left\{ x, \mathbb {E}\left[\xi \right]\right\} \leqslant \mathbb {E}\left[\max \left\{ x, \xi \right\} \right] .

?

Problem 3.104

Let $\xi_{1}$ and $\xi_{2}$ be random variables with distribution functions $F_{1}(x)$ and $F_{2}(x)$ respectively. Prove that

\int _{x}^{\infty }\left(1-F_{1}(t)\right) d t \leqslant \int _{x}^{\infty }\left(1-F_{2}(t)\right) d t

if and only if

\mathbb {E}\left[\max \left\{ x, \xi _{1}\right\} \right] \leqslant \mathbb {E}\left[\max \left\{ x, \xi _{2}\right\} \right] .

?

Problem 3.105

Let $\xi$ and $\eta$ be random variables with zero mathematical expectations and such that for any $x$

\mathbb {E}\left[\max \left\{ x, \xi \right\} \right] \leqslant \mathbb {E}\left[\max \left\{ x, \eta \right\} \right] .

Prove that for any convex function $f(x)$

\mathbb {E}\left[f(\xi )\right] \leqslant \mathbb {E}\left[f(\eta )\right] .

?

Problem 3.106

Let $\xi_{1}$ and $\xi_{2}$ be random variables with distribution functions $F_{1}(x)$ and $F_{2}(x)$ respectively. Prove that if for any $x$

\int _{x}^{\infty }\left(1-F_{1}(t)\right) d t \leqslant \int _{x}^{\infty }\left(1-F_{2}(t)\right) d t,

then

\mathbb {E}\left[\xi _{1}^{r}\right] \leqslant \mathbb {E}\left[\xi _{2}^{r}\right]

for any $r \geqslant 1$ (provided that these moments exist).

?

Problem 3.107

Let $\xi$ be a random variable with finite absolute moment of order $\alpha > 0$ . Prove that

\mathbb {E}\left[\left|\xi +a\right|^{\alpha }\right] \rightarrow \infty

as $a \rightarrow \infty$ .

?

Problem 3.108

Let $\xi$ be a random variable with finite absolute moment of order $2 n-1$ ( $n$ a positive integer). Prove that there exists a real $a$ such that

\mathbb {E}\left[(\xi -a)^{2 n-1}\right] = 0,

and moreover such $a$ is unique.

?

Problem 3.109

Let $\xi$ be a random variable with finite absolute moment of order $\alpha > 0: \mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty$ . Prove that for any real $a$

\mathbb {E}\left[\left|\xi -a\right|^{\alpha }\right] < \infty .

?

Problem 3.110

Prove that

\operatorname {Var}\left[\xi \right] = \min _{a} \mathbb {E}\left[(\xi -a)^{2}\right] .

?

Problem 3.111

Let $\xi$ be a random variable with finite absolute moment of order $2 n: \mathbb {E}\left[\xi^{2 n}\right] < \infty$ . Prove that a number $a$ satisfies the condition

\mathbb {E}\left[(\xi -a)^{2 n-1}\right] = 0

if and only if

\mathbb {E}\left[(\xi -a)^{2 n}\right] < \mathbb {E}\left[(\xi -b)^{2 n}\right]

for any real $b$ .

?

Problem 3.112

Let $\xi$ and $\eta$ be random variables with distribution densities $f(x)$ and $g(x)$ respectively, and suppose there exists an $a$ such that $f(x) \leqslant g(x)$ for $x > a$ and $f(x) \geqslant g(x)$ for $x < a$ . Prove that $\mathbb {E}\left[\xi \right] \leqslant \mathbb {E}\left[\eta \right]$ , if the indicated mathematical expectations exist, and if in addition $f(x) = g(x) = 0$ for $x < 0$ , then $\mathbb {E}\left[\xi^{\alpha }\right] \leqslant \mathbb {E}\left[\eta^{\alpha }\right]$ for any $\alpha > 0$ .

?

Problem 3.113

Let $F_{1}(x)$ and $F_{2}(x)$ be distribution functions, $a_{1}$ and $a_{2}$ the corresponding mathematical expectations (assumed to exist). Prove that if $F_{1}(x) \geqslant F_{2}(x)$ for any $x$ , then $a_{1} \leqslant a_{2}$ .

?

Problem 3.114

Let $\xi_{1}$ and $\xi_{2}$ be random variables with distribution densities $p_{1}(x)$ and $p_{2}(x)$ , symmetric about zero, and finite variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ respectively, with $p_{1}(x) = p_{2}(x) = 0$ for $\left|x\right| > a(a > 0)$ . Let $p_{1}(x)$ be convex from below and $p_{2}(x)$ convex from above on the interval $[-a, a]$ . Which is larger: $\sigma_{1}^{2}$ or $\sigma_{2}^{2}$ ?

?

Problem 3.115

Can it be said, in the previous problem, which is larger, $\sigma_{1}^{2}$ or $\sigma_{2}^{2}$ , if the assumption of symmetry of the distributions of the random variables $\xi_{1}$ and $\xi_{2}$ is dropped?

?

Problem 3.116

Prove the Cauchy-Bunyakovsky-Schwarz inequality.

?

Problem 3.117

(Hölder's inequality) Prove that if $r > 1$ and $\frac{1}{r}+\frac{1}{s} = 1$ , then

\mathbb {E}\left[\left|\xi \eta \right|\right] \leqslant \left(\mathbb {E}\left[\left|\xi \right|^{r}\right]\right)^{1 / r}\left(\mathbb {E}\left[\left|\eta \right|^{s}\right]\right)^{1 / s} .

?

Problem 3.118

Prove that $\left(\mathbb {E}\left[\left|\xi \right|^{r}\right]\right)^{1 / r}$ is a nondecreasing function of $r$ .

?

Problem 3.119

Prove that for any random variables $\xi_{1}, \ldots , \xi_{n}$ with finite moments of order $\alpha \geqslant 1$ the inequality

\mathbb {E}\left[\left|\xi _{1}+\ldots +\xi _{n}\right|^{\alpha }\right] \leqslant n^{\alpha -1}\left(\mathbb {E}\left[\left|\xi _{1}\right|^{\alpha }\right]+\ldots +\mathbb {E}\left[\left|\xi _{n}\right|^{\alpha }\right]\right) .

?

Problem 3.120

Prove that for any random variables $\xi_{i}, \ldots , \xi_{n}$ having finite moments of order $\alpha \leqslant 1$ , the inequality

\mathbb {E}\left[\left|\xi _{1}+\ldots +\xi _{n}\right|^{\alpha }\right] \leqslant \mathbb {E}\left[\left|\xi _{1}\right|^{\alpha }\right]+\ldots +\mathbb {E}\left[\left|\xi _{n}\right|^{\alpha }\right] .

holds.

?

Problem 3.121

(Minkowski's inequality) Prove that for $r \geqslant 1$

\left(\mathbb {E}\left[\left|\xi +\eta \right|^{r}\right]\right)^{1 / r} \leqslant \left(\mathbb {E}\left[\left|\xi \right|^{r}\right]\right)^{1 / r}+\left(\mathbb {E}\left[\left|\eta \right|^{r}\right]\right)^{1 / r} .

?

Problem 3.122

Prove that $\log \mathbb {E}\left[\left|\xi \right|^{r}\right]$ is a convex function of $r(r \geqslant 0)$ .

?

Problem 3.123

Prove that $\log \mathbb {E}\left[\left|\xi \right|^{r}\right]$ , as a function of $r(r \geqslant 0)$ , is linear if and only if $\xi$ has a degenerate distribution.

?

Problem 3.124

Let $\xi$ be a random variable. Set $\beta_{k} = \mathbb {E}\left[\left|\xi \right|^{k}\right]$ . Prove that if $0 \leqslant l \leqslant m \leqslant n$ , then $\beta_{m}^{n-l} \leqslant \beta_{l}^{n-m} \beta_{n}^{m-l}$ .

?

Problem 3.125

Let $\xi$ be a random variable. Prove that for $0 < r \leqslant 1$ , for any real $a$

\frac{1}{2} \mathbb {E}\left[\left|\xi -m \xi \right|^{r}\right] \leqslant \mathbb {E}\left[\left|\xi ^{(s)}\right|^{r}\right] \leqslant 2 \mathbb {E}\left[\left|\xi -a\right|^{r}\right] .

?

Problem 3.126

Let $\xi$ be a random variable. Prove that for $1 < r < 2$ , for any real $a$

\frac{1}{2} \mathbb {E}\left[\left|\xi -m \xi \right|^{r}\right] \leqslant \mathbb {E}\left[\left|\xi ^{(s)}\right|^{r}\right] \leqslant 2^{r} \mathbb {E}\left[\left|\xi -a\right|^{r}\right]

?

Problem 3.127

Let $\xi$ and $\eta$ be independent random variables, with $\eta$ having a symmetric distribution. Prove that for any $1 \leqslant \alpha \leqslant 2$

\mathbb {E}\left[\left|\xi +\eta \right|^{\alpha }\right] \leqslant \mathbb {E}\left[\left|\xi \right|^{\alpha }\right]+\mathbb {E}\left[\left|\eta \right|^{\alpha }\right] .

?

Problem 3.128

Prove that if the random variables $\xi_{1}, \ldots , \xi_{n}$ are independent and have symmetric distributions, then for any $1 \leqslant \alpha \leqslant 2$

\mathbb {E}\left[\left|\xi _{1}+\ldots +\xi _{n}\right|^{\alpha }\right] \leqslant \mathbb {E}\left[\left|\xi _{1}\right|^{\alpha }\right]+\ldots +\mathbb {E}\left[\left|\xi _{n}\right|^{\alpha }\right] .

?

Problem 3.129

Prove that if the random variables $\xi$ and $\eta$ are independent and $\mathbb {E}\left[\eta \right] = 0, \mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty , \mathbb {E}\left[\left|\eta \right|^{\alpha }\right] < \infty , \alpha \geqslant 1$ , then

\mathbb {E}\left[\left|\xi +\eta \right|^{\alpha }\right] \geqslant \mathbb {E}\left[\left|\xi \right|^{\alpha }\right] .

?

Problem 3.130

Prove that if $\xi_{1}, \ldots , \xi_{n}$ are independent random variables, $\mathbb {E}\left[\xi_{i}\right] = 0, \mathbb {E}\left[\left|\xi_{i}\right|^{\alpha }\right] < \infty , 1 \leqslant \alpha \leqslant 2$ , then

\mathbb {E}\left[\left|\xi _{1}+\ldots +\xi _{n}\right|^{\alpha }\right] \leqslant 2^{\alpha }\left(\mathbb {E}\left[\left|\xi _{1}\right|^{\alpha }\right]+\ldots +\mathbb {E}\left[\left|\xi _{n}\right|^{\alpha }\right]\right) .

?

Problem 3.131

The random variable $\xi$ has a lattice distribution with step $h$ . Set

v_{h} = \mathbb {E}\left[\left|\xi -\mathbb {E}\left[\xi \right]\right|^{h}\right]

( $k$ a positive integer). Prove that

v_{k-1} \leqslant \frac{2}{h} v_{k}

?

Problem 3.132

Let $\xi$ be a positive nondegenerate random variable with finite mathematical expectation. Prove that

\frac{1}{\mathbb {E}\left[\xi \right]} \leqslant \mathbb {E}\left[\frac{1}{\xi }\right]

?

Problem 3.133

Let $\xi$ and $\eta$ be independent random variables taking positive values. Prove that for any $r \geqslant 0$

\mathbb {E}\left[\frac{\xi }{\eta }^{r}\right] \geqslant \frac{\mathbb {E}\left[\xi ^{r}\right]}{\mathbb {E}\left[\eta ^{r}\right]}

?

Problem 3.134

Find the correlation coefficient between the number of "aces" (ones) and the number of "sixes" that come up in $n$ independent throws of a fair die.

?

Problem 3.135

Let the nonnegative integer-valued random variables $\xi_{1}, \xi_{2}, \ldots , \xi_{s}$ be such that $\xi_{1}+\xi_{2}+\ldots +\xi_{s} = n$ and for any $m_{1}, \ldots , m_{s}, m_{1} \geqslant 0, m_{1}+\ldots +m_{s} = n$ ,

\mathbb {P}\left(\xi _{1} = m_{1}, \ldots , \xi _{s} = m_{s}\right) = \frac{n!}{m_{1}!\ldots m_{s}!} p_{1}^{m_{1}} \ldots p_{s}^{m_{s}}

where $p_{i} \geqslant 0, p_{1}+\ldots +p_{s} = 1$ . Find the correlation coefficient between $\xi_{i}$ and $\xi_{j}, i, j = 1,2, \ldots , s$ .

?

Problem 3.136

Prove that the correlation coefficient of any two random variables having finite nonzero variances lies between $-1$ and $+1$ ; equals zero if the variables are independent; equals $-1$ or $+1$ if and only if the random variables are linearly related.

?

Problem 3.137

Construct an example showing that the vanishing of the correlation coefficient of two random variables does not imply their independence.

?

Problem 3.138

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables with finite nonzero variances. Prove that $D\left(\xi_{1}+\ldots +\xi_{n}\right) = \operatorname {Var}\left[\xi_{1}\right]+\ldots +\operatorname {Var}\left[\xi_{n}\right]$ if and only if $\xi_{i}$ and $\xi_{j}, \ldots$ , are pairwise uncorrelated.

?

Problem 3.139

The random variables $\xi_{1}, \ldots , \xi_{n+m}(n \geqslant m)$ are independent, identically distributed and have finite variance. Find the correlation coefficient between the random variables $\eta_{1} = \xi_{1}+\ldots +\xi_{n}$ and $\eta_{2} = \xi_{m+1}+\ldots +\xi_{m+n}$ .

?

Problem 3.140

The random variables $\xi$ and $\eta$ are independent and have the same distribution with mathematical expectation $a$ and variance $\sigma^{2}$ . Find the correlation coefficient of the random variables $\xi_{1} = \alpha \xi +\beta \eta$ and $\xi_{2} = \alpha \xi -\beta \eta$ .

?

Problem 3.141

Let the joint distribution of the random variables $\xi$ and $\eta$ be normal, with $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\eta \right] = 0$ , and let the correlation coefficient of $\xi$ and $\eta$ equal $\rho$ . Find the correlation coefficient of the random variables $\xi^{2}$ and $\eta^{2}$ .

?

Problem 3.142

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables, with the correlation coefficient of any two of them equal to $\rho$ . Prove that $\rho \geqslant -1 /(n-1)$ .

?

Problem 3.143

The random variables $\xi$ and $\eta$ are independent,

\begin{gathered} \mathbb {P}\left(\xi = 1\right) = \mathbb {P}\left(\xi = -1\right) = 1 / 2, \quad \mathbb {P}\left(\eta = 1\right) = \mathbb {P}\left(\eta = -1\right) = 1 / 4, \\ \mathbb {P}\left(\eta = 0\right) = 1 / 2. \end{gathered}

Will the random variables $\xi \eta$ and $\eta$ be independent?

?

Problem 3.144

The random variables $\xi$ and $\eta$ are independent, $\mathbb {P}\left(\xi = 1\right) = \mathbb {P}\left(\xi = -1\right) = 1 / 2$ , $\mathbb {P}\left(\eta = 1\right) = \mathbb {P}\left(\eta = -1\right) = 1 / 4$ , $\mathbb {P}\left(\eta = 0\right) = 1 / 2$ (see problem 3.143). Under these conditions, will the random variables $\xi \eta$ and $\eta$ be uncorrelated?

?

Problem 3.145

Let $\xi$ be a random variable with finite variance. Prove that the correlation coefficient of the random variables $\xi$ and $\operatorname {sign} \xi$ is nonnegative.

?

Problem 3.146

Let $\xi$ be a random variable with a symmetric distribution and finite variance. Find the correlation coefficient of the random variables $\xi$ and $\left|\xi \right|$ .

?

Problem 3.147

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables, and let $\sigma_{i j}$ be the covariance between $\xi_{i}$ and $\xi_{j}(i, j = 1, \ldots , n)$ . Prove that the covariance matrix

\left[\begin{smallmatrix} \sigma _{11} & \cdots & \sigma _{1 n} \\ \cdots & \cdots & . \\ \sigma _{n 1} & \cdots & \sigma _{n n} \end{smallmatrix}\right]

is nonnegative definite.

?

Problem 3.148

Let $\xi$ be a random variable.

\mathbb {P}\left(\xi > 0\right) = \alpha , \quad \mathbb {P}\left(\xi < 0\right) = \beta , \quad \mathbb {E}\left[\xi \right] = a, \quad \mathbb {E}\left[\left|\xi \right|\right] = b .

Find $\operatorname {Cov}\left[ \xi , \operatorname {sign} \xi \right]$ .

?

Problem 3.149

Let the random variables $\xi_{1}$ and $\xi_{2}$ be independent, identically distributed, and have finite variances. Prove that the random variables $\eta_{1} = \xi_{1}+\xi_{2}$ and $\eta_{2} = \xi_{1}-\xi_{2}$ are uncorrelated. Can it be asserted that they are independent?

?

Problem 3.150

Let $\xi$ and $\eta$ be random variables with zero mathematical expectations, unit variances, and correlation coefficient $\rho$ . Prove that

\mathbb {E}\left[\max \left\{ \xi ^{2}, \eta ^{2}\right\} \right] \leqslant 1+\sqrt{1-\rho ^{2}} .

?

Problem 3.151

(a)(b)

Let $\xi , \eta$ and $\zeta$ be pairwise uncorrelated random variables. Can it be asserted that the following random variables will be uncorrelated:

?

(a)

$\xi$ and $\eta +\xi$ ;

(b)

$\xi$ and $\eta \xi$ ?

Problem 3.152

Let $\xi , \eta$ and $\zeta$ be independent random variables with finite positive variances. Can the random variables $\xi +\zeta$ and $\eta +\zeta$ be independent?

?

Problem 3.153

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables with finite third moment, with $\mathbb {E}\left[\xi_{i}-\mathbb {E}\left[\xi_{i}\right]^{3}\right] = 0$ . Find the correlation coefficient of the random variables

\xi = \frac{1}{n} \sum _{i = 1}^{n} \xi _{i} \text{ and } s^{2} = \frac{1}{n} \sum _{i = 1}^{n}\left(\xi _{i}-\bar{\xi }\right)^{2} .

?

Problem 3.154

Prove that if each of the independent random variables $\xi_{1}$ and $\xi_{2}$ has a geometric distribution, then the random variable $\eta = \min \left\{ \xi_{1}, \xi_{2}\right\}$ also has a geometric distribution. Find the parameter of this distribution if the parameters of the distributions of $\xi_{1}$ and $\xi_{2}$ equal $p_{1}$ and $p_{2}$ respectively.

?

Problem 3.155

The random variable $\xi$ takes nonnegative integer values. Prove that the following assertions are equivalent:

$\xi$ has a geometric distribution;
$\mathbb {P}\left( \xi = n+k \mid \xi \geqslant k \right) = \mathbb {P}\left(\xi = n\right), \quad k, n = 0,1,2, \ldots$

?

Problem 3.156

The random variables $\xi_{1}$ and $\xi_{2}$ are independent and have one and the same geometric distribution. Prove that

\mathbb {P}\left( \xi _{1} = k \mid \xi _{1}+\xi _{2} = n \right) = \frac{1}{n+1}, \quad k = 0,1, \ldots , n .

?

Problem 3.157

The random variable $\xi$ has a geometric distribution with parameter $p$ . Find the distribution of the random variable

\eta = \frac{\xi }{2}\left(1-(-1)^{\xi }\right) .

?

Problem 3.158

The sum of two independent integer-valued nonnegative random variables has a binomial distribution. Prove that each summand has a binomial distribution.

?

Problem 3.159

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, each of which takes the value 1 with probability $p$ and the value 0 with probability $1-p$ , and let $v$ be a nonnegative integer-valued random variable independent of $\xi_{1}, \xi_{2}, \ldots$ . Prove that the random variables $\xi_{1}+\ldots +\xi_{v}$ and $v-\left(\xi_{1}+\ldots +\xi_{v}\right)$ are independent if and only if $v$ has a Poisson distribution.

?

Problem 3.160

The random variables $\xi_{1}$ and $\xi_{2}$ have Poisson distributions with parameters $\lambda_{1}$ and $\lambda_{2}$ respectively, with $\lambda_{1} \leqslant \lambda_{2}$ . Prove that for any $t \geqslant 0$

\mathbb {P}\left(\xi _{1} \leqslant t\right) \geqslant \mathbb {P}\left(\xi _{2} \leqslant t\right) .

?

Problem 3.161

The random variables $\xi_{1}$ and $\xi_{2}$ are independent and have Poisson distributions with parameters $\lambda_{1}$ and $\lambda_{2}$ respectively. Find

\mathbb {P}\left( \xi _{1} = k \mid \xi _{1}+\xi _{2} = n \right), \quad k = 0,1, \ldots , n .

?

Problem 3.162

The random variables $\xi_{1}$ and $\xi_{2}$ are normally distributed with parameters $\left(0, \sigma_{1}^{2}\right)$ and $\left(0, \sigma_{2}^{2}\right)$ respectively. Prove that if $\sigma_{1}^{2} \geqslant \sigma_{2}^{2}$ , then

\mathbb {P}\left(\left|\xi _{1}\right| \leqslant t\right) \leqslant \mathbb {P}\left(\left|\xi _{2}\right| \leqslant t\right)

for any $t \geqslant 0$ .

?

Problem 3.163

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables having the same normal distribution. Prove that the random variables $\xi_{1}+\xi_{2}$ and $\xi_{1}-\xi_{2}$ are independent.

?

Problem 3.164

The random variable $\xi$ has a normal distribution with mathematical expectation $a$ and variance $\sigma^{2}$ . Find the distribution of the random variable $\operatorname {sign} \xi$ .

?

Problem 3.165

Let $\xi$ and $\eta$ be independent random variables having the same normal distribution with zero mathematical expectation and unit variance. Find the distribution of the random variable $\xi^{2}+\eta^{2}$ .

?

Problem 3.166

(Continuation) Under the same conditions, find the distribution of the random variable $\sqrt{\xi^{2}+\eta^{2}}$ .

?

Problem 3.167

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables having a normal distribution with zero mathematical expectation and unit variance. Find the distribution of the random variable $\sqrt{\xi_{1}^{2}+\ldots +\xi_{n}^{2}}$ .

?

Problem 3.168

(Continuation) Under the same conditions, find the distribution of the random variable $\xi_{1}^{2}+\ldots +\xi_{n}^{2}$ (the distribution of this random variable is called the chi-squared distribution with $n$ degrees of freedom).

?

Problem 3.169

Let $P(x)$ and $Q(x)$ be normal distribution functions with mathematical expectations $a$ and $b$ and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ respectively. Choose a distribution $R$ so that the equality $P * R = Q$ holds. What conditions on the parameters of the distributions $P$ and $Q$ must be imposed for such a distribution to exist?

?

Problem 3.170

Let $P$ be the uniform distribution on the interval $[0,1]$ , $Q$ the distribution assigning the points $-\pi$ and $-\pi +1 / 2$ each a probability of $1 / 2$ . Find the distribution $R$ satisfying the relation $R * Q = P$ .

?

Problem 3.171

Let $\xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random variables having the uniform distribution on the interval $[0,1]$ . Find the density of the distribution of the random variable

\eta _{n} = \prod _{k = 0}^{n} \xi _{k}

?

Problem 3.172

Let $\xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having the uniform distribution on the interval $[0,1]$ . Find the distribution of the random variable

\eta = \min \left\{ n: \prod _{k = 0}^{n} \xi _{k} < e^{-\lambda }\right\} .

?

Problem 3.173

Let $\xi$ be a random variable having the uniform distribution on the interval $[0,1]$ . Find the distribution of the random variable

-\frac{1}{\lambda } \log \xi , \quad \lambda > 0

?

Problem 3.174

The random variable $\xi$ has an exponential distribution with parameter 1. Find the distribution of the random variable $e^{-\xi }$ .

?

Problem 3.175

The random variables $\xi_{1}, \ldots , \xi_{n}$ are independent and have the same exponential distribution with parameter $\lambda$ . Prove that the random variables

\max \left\{ \xi _{1}, \ldots , \xi _{n}\right\} \text{ and } \xi _{1}+\frac{\xi _{2}}{2}+\frac{\xi _{3}}{3}+\ldots +\frac{\xi _{n}}{n}

are identically distributed.

?

Problem 3.176

Let $F(x)$ be a distribution function, $F(0) = 0$ and $F(x) < 1$ for some $x > 0$ . Prove that $F(x)$ is an exponential distribution function if and only if

F(x+y)-F(y) = F(x)(1-F(y))

for all $x, y \geqslant 0$ .

?

Problem 3.177

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables having the same normal distribution with zero mathematical expectation and unit variance. Find the distribution of the random variable $\xi_{1} / \xi_{2}$ .

?

Problem 3.178

Let $\xi$ be a random variable having the Cauchy distribution with density

\frac{b}{\pi \left((x-a)^{2}+b^{2}\right)}, \quad -\infty < x < \infty .

Find the density of the distribution of the random variable $1 / \xi$ .

?

Problem 3.179

The random variable $\xi$ has the uniform distribution on the interval $[0,1]$ . Find the density of the distribution of the random variable

\eta = \ln \frac{\xi }{1-\xi } .

?

Problem 3.180

The random variable $\xi$ has a logistic distribution with density

\frac{e^{x}}{\left(1+e^{x}\right)^{2}}, \quad -\infty < x < \infty .

Find the mathematical expectation and median of $\xi$ .

?

Problem 3.181

The random variable $\xi$ has the uniform distribution on the interval $[0,1]$ . Find the distribution of the random variable

\left(\frac{1}{\xi }\right)^{\alpha } \quad \alpha > 0.

?

Problem 3.182

Let $\xi$ and $\eta$ be independent random variables, with $\xi$ having an exponential distribution with parameter $\lambda$ , and $\eta$ uniformly distributed on the interval $[0, h]$ . Find the densities of the distributions of the random variables $\xi +\eta$ and $\xi -\eta$ respectively.

?

Problem 3.183

Let $\xi$ and $\eta$ be independent random variables, with $\xi$ uniformly distributed on the interval $[0,1]$ , and $\eta$ taking the values $0, \pm 1, \pm 2, \ldots$ with probabilities $p_{0}, p_{-1}, p_{1}, \ldots$ respectively; $p_{0}+p_{-1}+p_{1}+\ldots = 1$ . Find the density of the distribution of the sum $\xi +\eta$ .

?

Problem 3.184

Find the density of the distribution of the sum $\xi +\eta$ of independent random variables $\xi$ and $\eta$ , if $\xi$ has the uniform distribution on the interval $[a, b]$ , and $\eta$ the uniform distribution on the interval $[c, d](a < b, c < d, b-a \leqslant d-c)$ .

?

Problem 3.185

Let $\xi$ and $\eta$ be independent random variables with the same distribution density

g(x) = \frac{1}{2} e^{-\left|x\right|} .

Find the density of the distribution of the sum $\xi +\eta$ .

?

Problem 3.186

(a)(b)

Find the probability that the function $x^{2}-2 a x+b$ has complex roots, if the coefficients $a$ and $b$ are independent identically distributed random variables having:

?

(a)

the uniform distribution on the interval $[0, h]$ ,

(b)

the exponential distribution with parameter $\alpha$ .

Problem 3.187

Let $\xi$ and $\eta$ be independent random variables such that the sum $\xi +\eta$ has a degenerate distribution. Prove that each of the random variables $\xi$ and $\eta$ has a degenerate distribution. Can this be asserted if $\xi$ and $\eta$ are dependent?

?

Problem 3.188

Let $\xi$ and $\eta$ be independent random variables, with the random variables $\xi +\eta$ and $\xi$ identically distributed. Find the distribution of the random variable $\eta$ .

?

Problem 3.189

How many throws of a fair die must be made so that, with probability $1 / 2$ , the sum of the points shown exceeds 780?

?

Problem 3.190

Prove that in the class of distributions, the convolution operation is commutative and associative.

?

Problem 3.191

Prove that the convolution of two discrete distributions is discrete.

?

Problem 3.192

Prove that the convolution of a continuous distribution function with any distribution function is continuous.

?

Problem 3.193

Prove that the convolution of an absolutely continuous distribution with any distribution is absolutely continuous.

?

Problem 3.194

Prove that the convolution of an $n$ -times differentiable distribution function with any distribution function is $n$ -times differentiable.

?

Problem 3.195

Prove that the convolution of two symmetric distributions is symmetric.

?

Problem 3.196

Show that the convolution of two unimodal distributions need not be a unimodal distribution.

?

Problem 3.197

Prove that the convolution of two symmetric unimodal distributions is unimodal.

?

Problem 3.198

Let $\xi$ and $\eta$ be independent random variables, of which at least one has a continuous distribution. Prove that $\mathbb {P}\left(\xi = \eta \right) = 0$ .

?

Problem 3.199

Let $\xi_{1}, \xi_{2}, \xi_{3}$ be independent random variables with symmetric distributions, and suppose that for some $M > 0$

\mathbb {P}\left(\left|\xi _{1}+\xi _{2}+\xi _{3}\right| \leqslant M\right) = \mathbf{1} .

Prove that

\mathbb {P}\left(\left|\xi _{1}\right|+\left|\xi _{2}\right|+\left|\xi _{3}\right| \leqslant M\right) = 1.

?

Problem 3.200

(a)(b)

For each random variable $\xi$ set

s(\xi ) = \inf \left\{ x: x \geqslant 0, \quad \mathbb {P}\left(\left|\xi \right| \leqslant x\right) = 1\right\} .

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with symmetric distributions. Prove that:

?

(a)

$s\left(\xi_{1}+\ldots +\xi_{n}\right) = s\left(\xi_{1}\right)+\ldots +s\left(\xi_{n}\right)$ ,

(b)

$s\left(\xi_{1}+\ldots +\xi_{n}\right) = s\left(\left|\xi_{1}\right|+\ldots +\left|\xi_{n}\right|\right)$ .

Problem 3.201

In the notation of the previous problem (for each random variable $\xi$ set $s(\xi ) = \inf \left\{ x: x \geqslant 0, \mathbb {P}\left(\left|\xi \right| \leqslant x\right) = 1\right\}$ ), prove that the equalities

s\left(\xi _{1}+\ldots +\xi _{n}\right) = s\left(\xi _{1}\right)+\ldots +s\left(\xi _{n}\right), \qquad s\left(\xi _{1}+\ldots +\xi _{n}\right) = s\left(\left|\xi _{1}\right|+\ldots +\left|\xi _{n}\right|\right)

hold if $\xi_{1}, \ldots , \xi_{n}$ are independent identically distributed random variables (not necessarily symmetric).

?

Problem 3.202

Let $\xi$ be a random variable with the uniform distribution on the interval $[0,1]$ , and let the random variable $\eta$ , independent of $\xi$ , take the values $1,2,3, \ldots$ with probabilities $p_{1}, p_{2}, p_{3}, \ldots$ respectively, with $p_{1}+p_{2}+\ldots = 1, \quad p_{i} \geqslant p_{i+1}, \quad i = 1,2, \ldots$ . Prove that the random variable $\xi +\eta$ has a unimodal distribution.

?

Problem 3.203

Let $\xi$ be a nonnegative random variable. Set

\xi ^{(x)} = \begin{cases} \xi , & \text{ at } \xi > x, \\ 0, & \text{ at } \xi \leqslant x, \end{cases} \quad x \geqslant 0.

Prove that if $\xi$ and $\eta$ are nonnegative random variables with finite mathematical expectations, then

\mathbb {E}\left[(\xi +\eta )^{(x)}\right] \leqslant 2\left(\mathbb {E}\left[\xi ^{\left(\frac{x}{2}\right)}\right]+\mathbb {E}\left[\eta ^{\left(\frac{x}{2}\right)}\right]\right) .

?

Problem 3.204

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables, $\eta = \xi_{1}+\ldots +\xi_{n}$ . Prove that if for some $c > 0$ , $\mathbb {P}\left(0 < \eta < c\right) = 1$ , then

\mathbb {P}\left(0 < \xi _{i} < c / n\right) = 1

?

Problem 3.205

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, with $\mathbb {P}\left(\xi_{i} = 0\right) < 1$ . Set $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $c > 0$ there exists $n = n(c)$ such that

\mathbb {P}\left(\left|\eta _{n}\right| > c\right) > 0

?

Problem 3.206

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed positive random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $c > 0$ there is an $\alpha = \alpha (c) > 0$ such that

\mathbb {P}\left(\eta _{n} \leqslant c\right) \leqslant e^{-\alpha n}

?

Problem 3.207

Let $\xi$ and $\eta$ be independent identically distributed random variables, $\mathbb {P}\left(\xi > 0\right) = \mathbb {P}\left(\eta > 0\right) \geqslant 1 / 2$ . Is it true that $\mathbb {P}\left(\xi +\eta > 0\right) \geqslant 1 / 2$ ?

?

Problem 3.208

Let $\xi$ be an integer-valued nonnegative random variable, $\mathbb {P}\left(\xi = 0\right) > 0$ , and let $\xi = \xi_{1}+\xi_{2}$ , where $\xi_{1}$ and $\xi_{2}$ are independent identically distributed random variables. Prove that $\xi_1$ and $\xi_2$ can take only integer values.

?

Problem 3.209

Let $\xi$ and $\eta$ be independent random variables with lattice distributions with steps $a$ and $b$ . Prove that the sum $\xi +\eta$ has a lattice distribution if and only if $a / b$ is a rational number.

?

Problem 3.210

Let $F(x), F_{1}(x), F_{2}(x)$ be distribution functions, with $F = F_{1} * F_{2}$ , and let $N, n, m$ be the number of discontinuity points of the functions $F(x), F_{1}(x)$ and $F_{2}(x)$ respectively. Prove that $n+m-1 \leqslant N \leqslant nm$ .

?

Problem 3.211

Let $\xi$ and $\eta$ be independent random variables with distribution densities $p(x)$ and $q(x)$ respectively. Prove that the density of the distribution of the sum $\xi +\eta$ equals

r(x) = \int _{-\infty }^{\infty } p(x-t) q(t) d t = \int _{-\infty }^{\infty } p(t) q(x-t) d t

?

Problem 3.212

Let $\xi$ and $\eta$ be independent random variables, with $\xi$ having a distribution density that vanishes nowhere. Prove that the density of the distribution of the sum $\xi +\eta$ also vanishes nowhere.

?

Problem 3.213

Let $\xi$ and $\eta$ be independent identically distributed integer-valued random variables, $p_{i} = \mathbb {P}\left(\xi = i\right), \quad i = 0, \pm 1, \pm 2, \ldots$ . Prove that

\mathbb {P}\left(\xi -\eta = 0\right) = \sum _{i = -\infty }^{\infty } p_{i}^{2}

?

Problem 3.214

Let $\xi$ and $\eta$ be independent random variables with the same distribution density $p(x)$ . Denote by $q(x)$ the density of the distribution of the random variable $\xi -\eta$ . Prove that

q(0) = \int _{-\infty }^{\infty } p^{2}(x) d x

?

Problem 3.215

Let $\xi_{1}, \ldots , \xi_{n}$ and $\eta_{1}, \ldots , \eta_{n}$ be two collections, independent within each collection, of random variables. Prove that if for any real $a$ and any $k = 1,2, \ldots , n$

\mathbb {P}\left(\xi _{k} \geqslant a\right) \geqslant \mathbb {P}\left(\eta _{k} \geqslant a\right),

then

\mathbb {P}\left(\xi _{1}+\ldots +\xi _{n} \geqslant a\right) \geqslant \mathbb {P}\left(\eta _{1}+\ldots +\eta _{n} \geqslant a\right) .

?

Problem 3.216

Let $\xi$ and $\eta$ be independent random variables. Prove that the concentration function of their sum does not exceed the concentration functions of the summands:

Q_{\xi +\eta }(x) \leqslant \min \left\{ Q_{\xi }(x), Q_{\eta }(x)\right\}

(for the definition of the concentration function, see problem 3.54).

?

Problem 3.217

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables. Prove that for any nonnegative $x_{1}$ and $x_{2}$

Q_{\xi _{1}}\left(x_{1}\right) Q_{\xi _{2}}\left(x_{2}\right) \leqslant Q_{\xi _{1}+\xi _{2}}\left(x_{1}+x_{2}\right) .

?

Problem 3.218

(I. V. Romanovsky's theorem) Let $\xi_{1}$ and $\xi_{2}$ be independent random variables, $\xi = \xi_{1}+\xi_{2}$ . Prove that for any $x \geqslant 0$

Q_{\xi }(x) \leqslant Q_{\xi _{1}}(x) Q_{\xi _{2}}(x)+\left(1-Q_{\xi _{1}}(x)\right)\left(1-Q_{\xi _{2}}(x)\right) .

?

Problem 3.219

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables, $\xi = \xi_{1}+\xi_{2}$ ; suppose that for some Borel set $A$

\mathbb {P}\left(\xi \in A\right) > 1-\varepsilon , \quad 0 < \varepsilon \leqslant 1

Prove that for some real $a$

\mathbb {P}\left(\xi _{1} \in A+a\right) > 1-\varepsilon .

?

Problem 3.220

Let $\xi$ and $\eta$ be independent random variables, with $\xi$ having distribution density $p(x)$ . Denote by $q(x)$ the density of the distribution of the sum $\xi +\eta$ . Prove that

\sup _{x} q(x) \leqslant \sup _{x} p(x) .

?

Problem 3.221

Give an example of a random variable $\xi$ such that its distribution function $F(x)$ is representable in the form $F = F_{1} * F_{2}$ , where $F_{1}(x)$ and $F_{2}(x)$ are some nondegenerate distribution functions, but there do not exist two independent random variables $\xi_{1}$ and $\xi_{2}$ such that $\xi_{i}$ has distribution function $F_{i}(x), i = 1,2$ , and $\xi = \xi_{1}+\xi_{2}$ .

?

Problem 3.222

Let $\xi$ be a random variable, $\mathbb {P}\left(0 < \xi < c\right) = 1, c > 0$ , $\mathbb {E}\left[\xi \right] = a, \operatorname {Var}\left[\xi \right] = b$ . Prove that if $b > a$ , then for $n \geqslant c$ , $\xi$ cannot be represented as the sum of $n$ independent identically distributed random variables.

?

Problem 3.223

Let $\xi$ be a random variable taking the values $0^{n}, 1^{n}, 2^{n}, \ldots (n \geqslant 2)$ with positive probabilities and

\sum _{i = 0}^{\infty } \mathbb {P}\left(\xi = i^{n}\right) = 1

Prove that $\xi$ cannot be represented as the sum of two independent random variables, each of which has a nondegenerate distribution.

?

Problem 3.224

Let the random variable $\xi$ take exactly two values. Prove that it cannot be represented as the sum of two independent random variables, each of which has a nondegenerate distribution.

?

Problem 3.225

The random variable $\xi$ takes the values $-1,0$ and 1 with probabilities $p_{1}, p_{2}$ and $p_{3}$ respectively ( $p_{1} p_{2} p_{3} > 0, p_{1}+p_{2}+p_{3} = 1$ ). What conditions must be imposed on $p_{1}, p_{2}$ and $p_{3}$ for the distribution of $\xi$ to be representable as the convolution of two identical distributions?

?

Problem 3.226

Let $\xi$ and $\eta$ be independent random variables, with the distribution function of $\xi$ strictly increasing. Prove that the distribution function of the sum $\xi +\eta$ is also strictly increasing.

?

Problem 3.227

Let $\xi$ and $\eta$ be independent integer-valued random variables. Prove that

\mathbb {P}\left(\xi +\eta \sqrt{2} = 0\right) = \mathbb {P}\left(\xi = 0\right) \mathbb {P}\left(\eta = 0\right) .

?

Problem 3.228

The real numbers $a_{1}, \ldots , a_{n}$ are called rationally independent if none of them is representable as a linear combination of the others with rational coefficients.

Let $a_{1}, \ldots , a_{n}$ be rationally independent numbers, and let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables taking integer values. Prove that for any real $b_{1}, \ldots , b_{n}$

\mathbb {P}\left(\sum _{i = 1}^{n} a_{i} \xi _{i} = 0\right) \leqslant \mathbb {P}\left(\sum _{i = 1}^{n} b_{i} \xi _{i} = 0\right) .

?

Problem 3.229

Let $\xi_{1}, \ldots , \xi_{n}$ be independent integer-valued random variables, and let $a_{1}, \ldots , a_{n}$ be rationally independent numbers. Prove that

\sup _{x} \mathbb {P}\left(\sum _{i = 1}^{\infty } a_{i} \xi _{i} = x\right) = \prod _{i = 1}^{n} \sup _{x_{i}} \mathbb {P}\left(\xi _{i} = x_{i}\right) .

?

Problem 3.230

(a)(b)

Let $\xi$ be an integer-valued random variable. Set

V(\xi ) = \sum _{i = -\infty }^{\infty }\left|\mathbb {P}\left(\xi = i\right)-\mathbb {P}\left(\xi = i+1\right)\right| .

?

(a)

Prove that $V(\xi ) \leqslant 2$ ,

(b)

let $\xi$ and $\eta$ be independent random variables. Prove that

V(\xi +\eta ) \leqslant \min \left\{ V(\xi ), V(\eta )\right\} .

Problem 3.231

Let $\xi$ be a nonnegative random variable with finite mathematical expectation. Prove that

\sum _{n = 1}^{\infty } \mathbb {P}\left(\xi \geqslant n\right) \leqslant \mathbb {E}\left[\xi \right] \leqslant 1+\sum _{n = 1}^{\infty } \mathbb {P}\left(\xi \geqslant n\right) .

?

Problem 3.232

Let $\xi$ be a random variable with finite variance. Prove that

\left|m \xi -\mathbb {E}\left[\xi \right]\right| \leqslant \sqrt{2 \operatorname {Var}\left[\xi \right]}

( $m \xi$ the median of $\xi$ ).

?

Problem 3.233

Let $\xi$ be a random variable with finite mathematical expectation. Prove that for any $\varepsilon > 0$

\mathbb {P}\left(\xi \geqslant \varepsilon \right) \leqslant \mathbb {E}\left[\max \left\{ 0, \xi \right\} \right] / \varepsilon .

?

Problem 3.234

Let $\xi$ be a nonnegative random variable, $\mathbb {E}\left[e^{h \xi }\right] < \infty$ , $h > 0$ . Prove that for any $\varepsilon > 0$

\mathbb {P}\left(\xi \geqslant \varepsilon \right) \leqslant \mathbb {E}\left[e^{h \xi }\right] / e^{h \xi } .

?

Problem 3.235

Let $\xi$ be a random variable, $f(x)$ a nonnegative nondecreasing function. Prove that for any real $c$

\mathbb {P}\left(\xi \geqslant c\right) \leqslant \mathbb {E}\left[f(\xi )\right] / f(c) .

?

Problem 3.236

Let $\xi$ be a random variable, $f(x)$ a positive function, nonincreasing for $x \geqslant 0$ . Prove that for any $a \geqslant 0$

\mathbb {P}\left(\xi \leqslant a\right) \leqslant \mathbb {E}\left[f(\xi )\right] / f(a)

?

Problem 3.237

Let $\xi$ be a random variable with finite mathematical expectation and finite variance. Prove that for any $x > 0$

\mathbb {P}\left(\xi \leqslant -x\right) \leqslant \frac{\operatorname {Var}\left[\xi \right]}{\operatorname {Var}\left[\xi \right]+x^{2}}, \quad \mathbb {P}\left(\xi \geqslant x\right) \leqslant \frac{\operatorname {Var}\left[\xi \right]}{\operatorname {Var}\left[\xi \right]+x^{2}} .

?

Problem 3.238

(Gauss's inequality) The random variable $\xi$ has a symmetric unimodal distribution. Prove that for any $\varepsilon \geqslant 0$

\mathbb {P}\left(\left|\xi \right| \geqslant \varepsilon \right) \leqslant \frac{4}{9} \frac{\operatorname {Var}\left[\xi \right]}{\varepsilon ^{2}} .

?

Problem 3.239

Let $\xi$ be a random variable having a unimodal distribution with vertex at the point $x_{0}$ and finite variance. Set

\tau ^{2} = \operatorname {Var}\left[\xi \right]+\left(x_{0}-\mathbb {E}\left[\xi \right]\right)^{2} .

Prove that for any $\varepsilon \geqslant 0$

\mathbb {P}\left(\left|\xi -x_{0}\right| \geqslant \varepsilon \tau \right) \leqslant \frac{4}{9 \tau ^{2}} .

?

Problem 3.240

Let the conditions of the previous problem hold. Set

s = \frac{\mathbb {E}\left[\xi \right]-x_{0}}{\sqrt{\operatorname {Var}\left[\xi \right]}} .

Prove that for any $\varepsilon > \left|s\right|$ the inequality

\mathbb {P}\left(\left|\xi -\mathbb {E}\left[\xi \right]\right| \geqslant \varepsilon \sqrt{\operatorname {Var}\left[\xi \right]}\right) \leqslant \frac{4\left(1+s^{2}\right)}{9(\varepsilon -\left|s\right|)^{2}} .

holds.

?

Problem 3.241

(a)(b)

Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random variables. Prove that:

?

(a)

$\frac{1}{2} \mathbb {P}\left(\xi_{1}-m \xi_{1} \geqslant \varepsilon \right) \leqslant \mathbb {P}\left(\xi_{1}-\xi_{2} \geqslant \varepsilon \right)$ ,

(b)

$\frac{1}{2} \mathbb {P}\left(\left|\xi_{1}-m \xi_{1}\right| \geqslant \varepsilon \right) \leqslant \mathbb {P}\left(\left|\xi_{1}-\xi_{2}\right| \geqslant \varepsilon \right)$ .

Problem 3.242

Prove that for any real $a$

\mathbb {P}\left(\left|\xi _{1}-\xi _{2}\right| \geqslant \varepsilon \right) \leqslant 2 \mathbb {P}\left(\left|\xi _{1}-a\right| \geqslant \varepsilon / 2\right) .

?

Problem 3.243

(a)(b)

Let $\xi$ and $\eta$ be independent random variables, with $\eta$ having zero median. Prove that for any $a$ :

?

(a)

$\mathbb {P}\left(\xi +\eta \geqslant a\right) \geqslant \frac{1}{2} \mathbb {P}\left(\xi \geqslant a\right)$ ,

(b)

$\mathbb {P}\left(\left|\xi +\eta \right| \geqslant a\right) \geqslant \frac{1}{2} \mathbb {P}\left(\left|\xi \right| \geqslant a\right)$ .

Problem 3.244

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables. Prove that for any nonnegative $x_{1}$ and $x_{2}$

\mathbb {P}\left(\left|\xi _{1}\right| \leqslant x_{1}\right) \mathbb {P}\left(\left|\xi _{2}\right| \leqslant x_{2}\right) \leqslant \mathbb {P}\left(\left|\xi _{1}+\xi _{2}\right| \leqslant x_{1}+x_{2}\right) .

?

Problem 3.245

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables with finite mathematical expectations, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $\varepsilon > 0$

\mathbb {P}\left(\max _{1 \leqslant k \leqslant n}\left|\eta _{k}\right| \geqslant \varepsilon \right) \leqslant \frac{\sum _{k = 1}^{n} \mathbb {E}\left[\left|\xi _{k}\right|\right]}{\varepsilon } .

?

Problem 3.246

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with zero mathematical expectations and finite absolute moments of order $k \geqslant 1$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $\varepsilon > 0$

\mathbb {P}\left(\max _{1 \leqslant j \leqslant n}\left|\eta _{j}\right| \geqslant \varepsilon \right) \leqslant \frac{\mathbb {E}\left[\left|\eta _{n}\right|^{k}\right]}{\varepsilon ^{k}}

(for $k = 2$ , Kolmogorov's inequality).

?

Problem 3.247

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with symmetric distributions, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any real $a$

\mathbb {P}\left(\max _{1 \leqslant h \leqslant n} \eta _{k} \geqslant a\right) \leqslant 2 \mathbb {P}\left(\eta _{n} \geqslant a\right) .

?

Problem 3.248

(a)(b)

(Lévy's inequalities) Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $\varepsilon > 0$ :

?

(a)

$\mathbb {P}\left(\max_{k \leqslant n}\left(\eta_{k}-m\left(\eta_{k}-\eta_{n}\right)\right) \geqslant \varepsilon \right) \leqslant 2 \mathbb {P}\left(\eta_{n} \geqslant \varepsilon \right)$ ;

(b)

$\mathbb {P}\left(\max_{k \leqslant n}\left|\eta_{k}-m\left(\eta_{k}-\eta_{n}\right)\right| \geqslant \varepsilon \right) \leqslant 2 \mathbb {P}\left(\left|\eta_{n}\right| \geqslant \varepsilon \right)$ ( $m(\cdot )$ the median).

Problem 3.249

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with zero mathematical expectations and finite variances, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any $\varepsilon \geqslant 0$

\mathbb {P}\left(\max _{1 \leqslant k \leqslant n} \eta _{k} \geqslant \varepsilon \right) \leqslant 2 \mathbb {P}\left(\eta _{n} \geqslant \varepsilon -\sqrt{2 \operatorname {Var}\left[\eta _{n}\right]}\right) .

?

Problem 3.250

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with symmetric distributions and finite absolute moments of order $r \geqslant 1$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that

\mathbb {E}\left[\max _{1 \leqslant k \leqslant n}\left|\eta _{k}\right|^{r}\right] \leqslant 2 \mathbb {E}\left[\left|\eta _{n}\right|^{r}\right] .

?

Problem 3.251

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with zero mathematical expectations and finite absolute moments of order $r \geqslant 1$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that

\mathbb {E}\left[\max _{1 \leqslant k \leqslant n}\left|\eta _{k}\right|^{r}\right] \leqslant 2^{2 r+1} \mathbb {E}\left[\left|\eta _{n}\right|^{r}\right] .

?

Problem 3.252

The function $\varphi (x)$ is even, nonnegative, and nonincreasing for $x \geqslant 0$ ; $\xi$ and $\eta$ are random variables. Prove that if for any $a \geqslant 0$

\mathbb {P}\left(\left|\xi \right| \leqslant a\right) \geqslant \mathbb {P}\left(\left|\eta \right| \leqslant a\right),

then

\mathbb {E}\left[\varphi (\xi )\right] \geqslant \mathbb {E}\left[\varphi (\eta )\right]

and conversely.

?

Problem 3.253

Prove the triangle inequality for the total variation distance, i.e., prove that for any three probability distributions $\mathbf{F}, \mathbf{G}$ and $\mathbb {P}$ , the inequality

\operatorname {Var}(\mathbf{F}, \mathbf{G}) \leqslant \operatorname {Var}(\mathbf{F}, \mathbb {P})+\operatorname {Var}(\mathbb {P}, \mathbf{G}) .

holds.

Prove the triangle inequality for the uniform distance $\sup_{x} \left|F(x)-G(x)\right|$ .

?

Problem 3.254

Let $\mathbb {P}$ and $\mathbf{Q}$ be absolutely continuous probability distributions, and let $p(x)$ and $q(x)$ be the corresponding distribution densities. Prove that

\operatorname {Var}(\mathbb {P}, \mathbf{Q}) = \frac{1}{2} \int _{-\infty }^{\infty }\left|p(x)-q(x)\right| d x .

?

Problem 3.255

Let $\xi$ and $\eta$ be random variables with distribution functions $F(x)$ and $G(x)$ respectively. Prove that

\sup _{x}\left|F(x)-G(x)\right| \leqslant \mathbb {P}\left(\xi \neq \eta \right) .

?

Problem 3.256

Find the uniform distance between two normal distributions with the same mathematical expectations and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ .

?

Problem 3.257

The uniform distance between the distributions of the random variables $\xi$ and $\eta$ equals $\rho$ . Find the uniform distance between the distributions of the random variables $c \xi$ and $c \eta \quad (c \neq 0)$ .

?

Problem 3.258

Find the uniform distance between two uniform distributions with the same mathematical expectations and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ .

?

Problem 3.259

Indicate a distribution equidistant, in the uniform metric, from two normal distributions with mathematical expectations $a_{1}$ and $a_{2}$ and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ respectively.

?

Problem 3.260

Let $F_{1}(x)$ and $F_{2}(x)$ be arbitrary distribution functions. Indicate a distribution function $F(x)$ equidistant, in the uniform metric, from the distribution functions $F_{1}(x)$ and $F_{2}(x)$ .

?

Problem 3.261

Find the uniform distribution on an interval equidistant, in the uniform metric, from two uniform distributions on the intervals $[-a, a]$ and $[-b, b]$ respectively.

?

Problem 3.262

Prove the triangle inequality for the Lévy distance.

?

Problem 3.263

Find the Lévy distance between two uniform distributions with zero mathematical expectations and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}\left(\sigma_{1} \leqslant \sigma_{2}\right)$ .

?

Problem 3.264

Let $\xi$ and $\eta$ be random variables with distribution functions $F(x)$ and $G(x)$ respectively. Prove that if

\mathbb {P}\left(\left|\xi -\eta \right| \geqslant \varepsilon \right) \leqslant \varepsilon

for some positive $\varepsilon > 0$ , then

L(F, G) \leqslant \varepsilon .

?

Problem 3.265

Let $\xi$ and $\eta$ be random variables with probability distributions $\mathbf{F}$ and $\mathbf{G}$ respectively. Prove that

\operatorname {Var}(\mathbf{F}, \mathbf{G}) \leqslant \mathbb {P}\left(\xi \neq \eta \right) .

?

Problem 3.266

Prove that for any distribution functions $F(x)$ and $G(x)$ the inequalities

L(F, G) \leqslant \sup _{x}\left|F(x)-G(x)\right| \leqslant (1+\beta ) L(F, G),

hold, where $\beta = \sup_{x} G^{\prime }(x)$ if $G(x)$ is differentiable, and $\beta = \infty$ otherwise.

?

Problem 3.267

Prove that for any two distribution functions $F(x)$ and $G(x)$

L^{2}(F, G) \leqslant \int _{-\infty }^{\infty }\left|F(x)-G(x)\right| d x

?

Problem 3.268

Let $F_{1}(x)$ and $F_{2}(x)$ be two distribution functions with the same mathematical expectations and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ respectively. Prove that

L\left(F_{1}, F_{2}\right) \leqslant 2\left(2 \max \left\{ \sigma _{1}, \sigma _{2}\right\} \right)^{2 / 3} .

?

Problem 3.269

Let $F(x)$ be a distribution function with zero mathematical expectation and unit variance, and let $\Phi (x)$ be the standard normal distribution function. Prove that

\sup _{x}\left|F(x)-\Phi (x)\right| \leqslant 0,5416.

?

Problem 3.270

Prove that for any three distribution functions $F_{1}(x), F_{2}(x)$ and $G(x)$ the inequality

\sup _{x}\left|F_{1} * G(x)-F_{2} * G(x)\right| \leqslant \sup _{x}\left|F_{1}(x)-F_{2}(x)\right| .

holds.

?

Problem 3.271

Prove that for any three probability distributions $\mathbf{F}_{1}, \mathbf{F}_{2}$ and $\mathbf{G}$ the inequality

\operatorname {Var}\left[\mathbf{F}_{1} * \mathbf{G}, \mathbf{F}_{2} * \mathbf{G}\right] \leqslant \operatorname {Var}\left[\mathbf{F}_{1}, \mathbf{F}_{2}\right] .

holds.

?

Problem 3.272

Prove that for any three distribution functions $F_{1}, F_{2}$ and $G$ the inequality

L\left(F_{1} * G, \quad F_{2} * G\right) \leqslant L\left(F_{1}, F_{2}\right) .

holds.

?

Problem 3.273

Let $F_{1}(x), \ldots , F_{n}(x), G_{1}(x), \ldots , G_{n}(x)$ be arbitrary distribution functions. Prove that

\sup _{x}\left|F_{1} * \ldots * F_{n}(x)-G_{1} * \ldots * G_{n}(x)\right| \leqslant \sum _{i = 1}^{n}\left(\sup _{x}\left|F_{i}(x)-G_{i}(x)\right|\right) .

?

Problem 3.274

Prove that for any probability distributions $\mathbf{F}_{1}(x), \ldots , \mathbf{F}_{n}(x), \mathbf{G}_{1}(x), \ldots , \mathbf{G}_{n}(x)$

\operatorname {Var}\left[\mathbf{F}_{1} * \ldots * \mathbf{F}_{n}, \mathbf{G}_{1} * \ldots * \mathbf{G}_{n}\right] \leqslant \sum _{i = 1}^{n} \operatorname {Var}\left[\mathbf{F}_{i}, \mathbf{G}_{i}\right] .

?

Problem 3.275

Prove that for any distribution functions $F_{1}(x), \ldots , F_{n}(x), \quad G_{1}(x), \ldots , G_{n}(x)$

L\left(F_{1} * \ldots * F_{n}, G_{1} * \ldots * G_{n}\right) \leqslant \sum _{i = 1}^{n} L\left(F_{i}, G_{i}\right)

?

Problem 3.276

Let $\xi$ and $\eta$ be random variables with distribution functions $F(x)$ and $G(x)$ respectively, and let $\zeta$ be a random variable, independent of $\xi$ and $\eta$ , with distribution function $H(x)$ . Prove that

\sup _{x}\left|F(x)-G(x)\right| \leqslant \sup _{x}\left|F * H(x)-G * H(x)\right|+\mathbb {P}\left(\zeta \neq 0\right) .

?

Problem 3.277

Prove that under the conditions of the previous problem

\operatorname {Var}(\mathbf{F}, \mathbf{G}) \leqslant \operatorname {Var}(\mathbf{F} * \mathbf{H}, \mathbf{G} * \mathbf{H})+\mathbb {P}\left(\zeta \neq 0\right) .

?

Problem 3.278

What conditions must the numbers $a, b$ and $c$ satisfy in order that, for a suitable choice of the normalizing factor $A$ , the function

A \exp \left\{ -\left(a x^{2}+2 b x+c y^{2}\right)\right\}

be a probability density on the plane?

?

Problem 3.279

Let $\xi$ be a random vector taking values in $\mathbb {R}^{n}$ . Prove that

\| \mathbb {E}\left[\xi \right]\| \leqslant \mathbb {E}\left[\| \xi \| \right] .

?

Problem 3.280

Show that the function

G(x, y) = \begin{cases} 1, & \text{ at } x+y \geqslant 0, \\ 0, & \text{ at } x+y < 0 \end{cases}

is continuous from the right, increasing in each variable, but is not a distribution function in $\mathbb {R}^{2}$ . Show the same for the function $G(x, y) = [x+y]$ (the integer part of $x+y$ ).

?

Problem 3.281

The random variables $\xi_{1}, \xi_{2}, \xi_{3}$ and $\xi_{4}$ are independent. Prove that the random vectors $\eta = \left(\xi_{1}, \xi_{2}\right)$ and $\zeta = \left(\xi_{3}, \xi_{4}\right)$ are independent. Is the converse true?

?

Problem 3.282

Give an example of a discontinuous two-dimensional probability density both of whose marginal densities are continuous.

?

Problem 3.283

Find and compare the marginal distributions of the uniform distribution in the unit square with the uniform distribution on its diagonal.

?

Problem 3.284

The random vector ( $\xi , \eta$ ) is uniformly distributed in a square with side equal to one, and diagonals coinciding with the coordinate axes. Find the correlation coefficient of the quantities $\xi$ and $\eta$ .

?

Problem 3.285

Let ( $\xi , \eta$ ) be a random vector whose distribution is concentrated on some line. Find the correlation coefficient of the quantities $\xi$ and $\eta$ .

?

Problem 3.286

Each of the random variables $\xi , \eta , \zeta$ has zero mathematical expectation and unit variance, and the relation

a \xi +b \eta +c \zeta = 0

holds ( $a, b$ and $c$ real numbers, $a b c \neq 0$ ). Find the covariance matrix of the random vector ( $\xi , \eta , \zeta$ ) and prove that

a^{4}+b^{4}+c^{4} \leqslant 2\left(a^{2} b^{2}+a^{2} c^{2}+b^{2} c^{2}\right)

?

Problem 3.287

The random vectors $\xi$ and $\eta$ are independent and each of them has the uniform distribution in a circle of unit radius centered at the origin. Find the density of the distribution of the sum $\xi +\eta$ .

?

Problem 3.288

Let $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ and $\eta = \left(\eta_{1}, \ldots , \eta_{n}\right)$ be independent random vectors taking values in $\mathbb {R}^{n}$ . Prove that for any $x, y \in \mathbb {R}^{n}$ the random variables $\langle \xi , x\rangle$ and $\langle \eta , y\rangle (\langle \cdot , \cdot \rangle$ the inner product $)$ are independent.

?

Problem 3.289

Let $\xi_{1}, \xi_{2}, \eta_{1}, \eta_{2}$ be random vectors taking values in $\mathbb {R}^{n}$ , where both $\xi_{1}$ and $\xi_{2}$ are independent of $\eta_{1}$ and $\eta_{2}$ . Will the random variables $\left\langle \xi_{1}, \xi_{2} \right\rangle$ and $\left\langle \eta_{1}, \eta_{2} \right\rangle$ be independent?

?

Problem 3.290

The random vector $\xi = \left(\xi_{1}, \xi_{2}\right)$ takes the values $(0,0)$ , $(0,1),(1,0),(1,1),(0,2),(1,2)$ , each with probability $1 / 6$ . Find the distribution of the random variable $\langle \xi , e\rangle$ , where $e$ is the vector with coordinates (1, 1).

?

Problem 3.291

The random vector $\xi = \left(\xi_{1}, \xi_{2}\right)$ has the uniform distribution in the triangle with vertices at the points $(-1,0),(0,1),(1,0)$ .

Find the distribution of the random variable $\langle \xi , e\rangle$ , where $e$ is the vector with coordinates (1/2, 1/2).

?

Problem 3.292

Let $0 < a \leqslant 1$ and

f(x, y) = ((1+a x)(1+a y)-a) e^{-x-y-a x y}

for $x > 0, y > 0$ and $f(x, y) = 0$ for other $x$ and $y$ . Prove that $f(x, y)$ is a two-dimensional probability density, and find its marginal distributions.

?

Problem 3.293

The correlation coefficient of the random variables $\xi$ and $\eta$ equals one. Can the random vector ( $\xi , \eta$ ) have a distribution density?

?

Problem 3.294

Let $\xi$ be a random variable with distribution function $F(x)$ . Find the distribution function $F_{2}(x, y)$ of the random vector ( $\xi , \xi$ ).

?

Problem 3.295

The random variable $\xi$ has a distribution density. Will the random vector $\left(\xi , \xi^{2}, \ldots , \xi^{n}\right)$ have a distribution density?

?

Problem 3.296

Let $\xi$ be a random variable with distribution function $F(x)$ . Find the distribution function $F_{2}(x, y)$ of the random vector $(\xi ,\left|\xi \right|)$ .

?

Problem 3.297

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with the same distribution function $F(x)$ . Set $\xi = \min \left\{ \xi_{1}, \ldots , \xi_{n}\right\}$ and $\eta = \max \left\{ \xi_{1}, \ldots , \xi_{n}\right\}$ . Find the distribution function of the random vector ( $\xi , \eta$ ).

?

Problem 3.298

Let $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ be a random vector taking values in $\mathbb {R}^{n}$ , and let $A$ be a nonrandom matrix with $m$ rows and $n$ columns. Prove that

\mathbb {E}\left[A \xi \right] = A(\mathbb {E}\left[\xi \right]) .

?

Problem 3.299

The random vector $\xi$ takes values in $\mathbb {R}^{n}$ and has a discrete distribution. Prove that there is a vector $e$ in $\mathbb {R}^{n}$ such that the distribution of $\xi$ is uniquely recovered from the distribution of the random variable $\langle \xi , e\rangle$ .

?

Problem 3.300

A probability distribution in $\mathbb {R}^{n}$ is called spherically symmetric if it is invariant under rotations about zero.

Let the random vectors $\xi$ and $\eta$ be independent and each of them have a spherically symmetric distribution. Prove that their sum $\xi +\eta$ also has a spherically symmetric distribution.

?

Problem 3.301

Let the random vector $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ have a spherically symmetric distribution. Prove that the random variables $\xi_{1}, \ldots , \xi_{n}$ are pairwise uncorrelated.

?

Problem 3.302

Prove that a two-dimensional distribution concentrated at $n$ points is completely determined by its projections onto $n+1$ pairwise noncollinear vectors.

?

Problem 3.303

Give an example of independent random vectors $\xi$ and $\eta$ , taking values in $\mathbb {R}^{n}$ , such that for any $a \in \mathbb {R}^{n}$ , $\xi +a$ and $\eta +a$ have distributions that are not spherically symmetric, but the distribution of their sum $\xi +\eta$ is spherically symmetric.

?

Problem 3.304

Let $\varphi_{1}(x, y)$ and $\varphi_{2}(x, y)$ be two normal two-dimensional distribution densities with the same mathematical expectations and variances, but different covariance matrices. Set

\psi (x, y) = \frac{1}{2}\left(\varphi _{1}(x, y)+\varphi _{2}(x, y)\right) .

Is the distribution density $\psi (x, y)$ normal?

?

Problem 3.305

(Continuation) Are the marginal distributions of the distribution with density $\psi (x, y)$ normal?

?

Problem 3.306

Let $u(x)$ be an odd continuous function on the line, equal to zero outside the interval $[-1,1]$ , with $\left|u(x)\right| < \frac{1}{\sqrt{2 \pi e}}$ . Prove that the function

f(x, y) = \frac{1}{2 \pi } e^{-\frac{x^{2}+y^{2}}{2}}+u(x) u(y)

is a two-dimensional distribution density which is not normal, but whose marginal distributions are normal.

?

Problem 3.307

Let $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ be a random vector taking values in $\mathbb {R}^{n}$ and having a normal distribution with covariance matrix

A =\left[\begin{smallmatrix} 1 & & & 0 \\ & \ddots & & \\ 0 & & & 1 \end{smallmatrix}\right]

Find the mathematical expectation and variance of the random variable $\langle \xi , e\rangle$ , where $e$ is an arbitrary unit vector in $\mathbb {R}^{n}$ .

?

Problem 3.308

The random variable $\xi$ has a normal distribution with zero mathematical expectation and unit variance. Find the distribution of the random vector

\eta = (\ln \left|\xi \right|, \operatorname {sign} \xi ) .

?

Problem 3.309

Prove that the $n$ -dimensional random vector $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ has a normal distribution if and only if, for any set of $n$ real numbers $c_{1}, \ldots , c_{n}$ , the linear combination

\sum _{i = 1}^{n} c_{i} \xi _{i}

has a normal distribution in $\mathbb {R}^{1}$ .

?

Problem 3.310

Prove that if the random vector $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ is normally distributed with zero mathematical expectation and unit covariance matrix, then the random point that is the endpoint of the vector $\xi /\| \xi \|$ is uniformly distributed on the unit sphere in $\mathbb {R}^{n}$ . Prove that the same holds for any random vector having a spherically symmetric distribution.

?

Problem 3.311

Let $\xi_{1}, \xi_{2}, \xi_{3}$ be independent identically distributed random variables,

\mathbb {E}\left[e^{i t \xi _{j}}\right] = f(t), \quad j = 1,2,3, \quad -\infty < t < \infty .

Find

\mathbb {E}\left[e^{i\left(u\left(\xi _{2}-\xi _{1}\right)+v\left(\xi _{3}-\xi _{1}\right)\right)}\right], \quad -\infty < u, v < \infty .

?

Problem 3.312

Prove that for a square matrix of size $n \times n$ to be the covariance matrix of some $n$ -dimensional probability distribution, it is necessary and sufficient that it be symmetric and nonnegative definite.

?

Problem 3.313

Let $\xi$ and $\eta$ be random variables with correlation coefficient $\rho$ . Prove the following two-dimensional analogue of Chebyshev's inequality:

\mathbb {P}\left(\left|\xi -\mathbb {E}\left[\xi \right]\right| \geqslant \varepsilon \sqrt{\operatorname {Var}\left[\xi \right]} \text{ or }\left|\eta -\mathbb {E}\left[\eta \right]\right| \geqslant \varepsilon \sqrt{\operatorname {Var}\left[\eta \right]}\right) \leqslant \frac{1}{\varepsilon ^{2}}\left(1+\sqrt{1-\rho ^{2}}\right) .

?

Problem 3.314

Let

A =\left[\begin{smallmatrix} \xi _{11}, \ldots , \xi _{1 n} \\ \cdots , \\ \xi _{n_{1}}, \ldots , \xi _{n n} \end{smallmatrix}\right], \quad n \geqslant 2,

be a matrix made up of random variables having the same finite mathematical expectation. Denote by $\left|A\right|$ the determinant of the matrix $A$ . Prove that if the rows of the matrix are independent, i.e., the random vectors $\xi_{1} = \left(\xi_{11}, \ldots , \xi_{1 n}\right), \ldots , \xi_{n} = \left(\xi_{n 1}, \ldots , \xi_{n n}\right)$ are independent, then the mathematical expectation of the determinant equals zero: $\mathbb {E}\left[\left|A\right|\right] = 0$ . Is the converse true? Will the assertion of the problem hold in the case when the elements of the matrix have different mathematical expectations?

?

Problem 3.315

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables, $\mathbb {P}\left(\xi_{i} = j\right) = 1 / N, j = 1,2, \ldots , N$ . Find the probability $\mathbb {P}\left(\xi_{1} = \ldots = \xi_{n}\right)$ .

?

Problem 3.316

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables taking the values $1,2, \ldots , N$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that

\mathbb {P}\left(\eta _{n} \text{ is divisible by } n\right) \geqslant 1 / N^{n-1} \text{. }

?

Problem 3.317

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

Let $a, b$ and $c$ be independent random variables, with $a$ and $b$ having the uniform distribution on the interval $[-1,1]$ , and $c$ taking the values $-1,0,1$ with probabilities $1 / 4,1 / 2,1 / 4$ respectively. Find the probability that the (random) function $f(x)$ is nondecreasing for $x \geqslant 0$ , if:

?

(a)

$f(x) = \frac{x}{2}+a \sin x ;$

(b)

$f(x) = \exp \left\{ a x-b x^{2}\right\}$ ;

(c)

$f(x) = \cos c x ;$

(d)

$f(x) = (a b+c) x$ .

Problem 3.318

Let $\xi$ be a random variable with median $m \xi$ . Prove that for any real $a$ , $m(a \xi ) = a m \xi$ .

?

Problem 3.319

Let $\xi_{1}$ and $\xi_{2}$ be random variables with distribution functions $F_{1}(x)$ and $F_{2}(x)$ respectively. Prove that if $F_{1}(x) \leqslant F_{2}(x)$ for all $x$ , then

\mathbb {E}\left[\max \left\{ x, \xi _{1}\right\} \right] \geqslant \mathbb {E}\left[\max \left\{ x, \xi _{2}\right\} \right] .

?

Problem 3.320

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with zero mathematical expectations and finite third moments. Prove that

\mathbb {E}\left[(\xi _{1}+\ldots +\xi _{n})^{3}\right] = \mathbb {E}\left[\xi _{1}^{3}\right]+\ldots +\mathbb {E}\left[\xi _{n}^{3}\right] .

?

Problem 3.321

Let $\xi_{1}$ and $\xi_{2}$ be independent random variables, $\xi = \xi_{1}+\xi_{2}, \lambda_{1}$ and $\lambda_{2}$ the medians of the distributions of $\xi_{1}$ and $\xi_{2}$ , and $y_{1}$ and $y_{2}$ respectively the lower and upper quartiles of the distribution of the random variable $\xi$ , i.e., $y_{1}$ is a number satisfying the condition

\mathbb {P}\left(\xi < y_{1}\right) \leqslant \frac{1}{4} \leqslant \mathbb {P}\left(\xi \leqslant y_{1}\right)

and $y_{2}$ is a number satisfying the condition

\mathbb {P}\left(\xi < y_{2}\right) \leqslant \frac{3}{4} \leqslant \mathbb {P}\left(\xi \leqslant y_{2}\right) .

Prove that

y_{1} \leqslant \lambda _{1}+\lambda _{2} \leqslant y_{2} .

?

Problem 3.322

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables with finite variance $\sigma^{2}$ . Set

\bar{\xi } = \frac{\xi _{1}+\ldots +\xi _{n}}{n} .

Find the mathematical expectation of the random variable

\frac{1}{n-1} \sum _{k = 1}^{n}\left(\xi _{k}-\bar{\xi }\right)^{2} .

?

Problem 3.323

Let $\xi$ be a random variable with finite variance. Prove that

\mathbb {P}\left(-3,2 < \frac{\xi -\mathbb {E}\left[\xi \right]}{\sqrt{\operatorname {Var}\left[\xi \right]}} < 3,2\right) > 0,9.

?

Problem 3.324

Let $\xi$ be a random variable and

a = \mathbb {E}\left[\xi \right], \quad 0 < b = \left(\mathbb {E}\left[(\xi -a)^{10}\right]\right)^{1 / 10} .

Prove that

\mathbb {P}\left(-2 < \frac{\xi -a}{b} < 2\right) > 0,999.

?

Problem 3.325

Let $\xi$ be a random variable with mathematical expectation $a$ and variance $\sigma^{2}$ . Prove that the probability that $\xi$ deviates from $a$ by more than $3 \sigma$ does not exceed 1/9.

?

Problem 3.326

Let $\xi$ and $\eta$ be random variables with finite second-order moments. Prove that

\mathbb {E}\left[(\eta -a \xi -b)^{2}\right] \geqslant \mathbb {E}\left[(\eta -a_{0} \xi -b_{0})^{2}\right] = \left(1-\rho ^{2}\right) \cdot \operatorname {Var}\left[\eta \right]

for any real $a$ and $b$ , where

a_{0} = \frac{\operatorname {Cov}\left[ \xi , \eta \right]}{\operatorname {Var}\left[\xi \right]}, \quad b_{0} = \mathbb {E}\left[\eta \right]-a_{0} \mathbb {E}\left[\xi \right], \quad \rho = \operatorname {Cov}\left[ \xi , \eta \right]

and $a_{0} = 0$ if $\operatorname {Var}\left[\xi \right] = 0$ .

?

Problem 3.327

Prove that the sequence of moments of any continuous distribution function $F(x)$ is positive definite, i.e., for any positive integer $m$ and any real $x_{0}, x_{1}, \ldots , x_{m}$

\sum _{i, k = 0}^{m} \alpha _{i+k} x_{i} x_{k} > 0,

where

\alpha _{l} = \int _{-\infty }^{\infty } x^{l} d F(x) .

?

Problem 3.328

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, each of which has the uniform distribution on the interval $[0,1]$ , and let $v$ be the random variable equal to that $k$ for which the sum $\eta_{k} = \xi_{1}+\ldots +\xi_{k}$ first exceeds one. Find $\mathbb {E}\left[v\right]$ .

?

Problem 3.329

Let $\xi$ be a random variable having a symmetric distribution and taking at least three values. Prove that on the same probability space there exist independent nondegenerate random variables $\eta$ and $\zeta$ such that $\xi = \eta \cdot \zeta$ .

?

Problem 3.330

Let $\xi$ and $\eta$ be random variables. Prove that if for every continuous bounded function $f(x)$

\mathbb {E}\left[f(\xi )\right] = \mathbb {E}\left[f(\eta )\right],

then $\xi$ and $\eta$ are identically distributed.

?

Problem 3.331

There are $n$ balls, among which $k$ are white and $n-k$ black. $v$ balls are chosen at random, where $v$ is a random variable taking the values from 1 to $n$ with equal probabilities. Find the mathematical expectation of the number of white balls among those chosen.

?

Problem 3.332

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables such that $\mathbb {P}\left(\xi_{i} = \xi_{j}\right) = 0$ for $i \neq j$ . Set $\zeta_{j i} = 1$ if $\xi_{j} \leqslant \xi_{i}$ , and $\zeta_{j i} = 0$ in the other cases, $i, j = 1,2, \ldots$ . Let

\eta _{i} = \sum _{j = 1}^{i} \zeta _{j i}, \quad i = 1,2, \ldots

Prove that $\eta_{1}, \eta_{2}, \ldots$ are independent random variables. Find the distribution of $\eta_{i}$ .

?

Problem 3.333

(a)(b)

Let $\xi$ and $\eta$ be random variables defined on the same probability space. The Ky Fan distance between $\xi$ and $\eta$ is the infimum of positive $\varepsilon$ such that $\mathbb {P}\left(\left|\xi -\eta \right| > \varepsilon \right) < \varepsilon$ .

On the probability space ( $\Omega , \mathscr {A}, P$ ), representing the interval $[0,1]$ with the $\sigma$ -algebra of Borel subsets as $\mathscr {A}$ and Lebesgue measure as $\mathbb {P}$ , two random variables $\xi$ and $\eta$ are given. Find the Ky Fan distance between them, if:

?

(a)

$\xi = \xi (\omega ) = \begin{cases} 1, & \text{ at } 0 \leqslant \omega \leqslant 1 / 2, \\ 0, & \text{ at } 1 / 2 < \omega \leqslant 1, \end{cases} \quad \eta = \eta (\omega ) = \begin{cases} 0, & \text{ at } 0 \leqslant \omega \leqslant 1 / 2, \\ 1, & \text{ at } 1 / 2 < \omega \leqslant 1. \end{cases}$

(b)

$\xi = \xi (\omega ) = \omega , \quad \eta = \eta (\omega ) = \omega / 2$ .

Problem 3.334

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite mathematical expectation, and let the random variable $\nu$ be independent of $\xi_{1}, \xi_{2}, \ldots$ and take positive integer values. Set $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that $\mathbb {E}\left[\eta_{\nu }\right] = \mathbb {E}\left[\xi_{1}\right] \cdot \mathbb {E}\left[\nu \right]$ .

?

Problem 3.335

Let

A =\left[\begin{smallmatrix} \xi _{11} & \ldots & \xi _{1 n} \\ \cdot & \cdot & \cdot \\ \xi _{n 1} & \ldots & \xi _{n n} \end{smallmatrix}\right]

be a matrix whose entries are independent random variables with zero mathematical expectations and the same variances $\sigma^{2}$ . Find the mathematical expectation and variance of the determinant of the matrix $A$ .

?

Problem 3.336

(Birnbaum's theorem) Let $\xi , \eta , \zeta$ be random variables, with $\zeta$ independent of $\xi$ and $\eta$ and having a symmetric unimodal distribution. Prove that if

\mathbb {P}\left(\left|\xi \right| \leqslant t\right) \geqslant \mathbb {P}\left(\left|\eta \right| \leqslant t\right)

for any $t \geqslant 0$ , then

\mathbb {P}\left(\left|\xi +\zeta \right| \leqslant t\right) \geqslant \mathbb {P}\left(\left|\eta +\zeta \right| \leqslant t\right) .

?

Problem 3.337

Let $\xi_{1}, \ldots , \xi_{n}$ and $\eta_{1}, \ldots , \eta_{n}$ be two collections, independent within each collection, of random variables having symmetric unimodal distributions. Prove that if

\mathbb {P}\left(\left|\xi _{k}\right| \leqslant t\right) \geqslant \mathbb {P}\left(\left|\eta _{k}\right| \leqslant t\right)

for any $t \geqslant 0$ and all $k = 1,2, \ldots , n$ , then

\mathbb {P}\left(\left|\sum _{k = 1}^{n} \xi _{k}\right| \leqslant t\right) \geqslant \mathbb {P}\left(\left|\sum _{k = 1}^{n} \eta _{k}\right| \leqslant t\right) .

?

Problem 3.338

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}, F_{n}(x)$ the distribution function of $\eta_{n}$ , $F\left(x_{1}, \ldots , x_{n}\right) = \mathbb {P}\left(\eta_{1} \leqslant x_{1}, \ldots , \eta_{n} \leqslant x_{n}\right)$ . Prove that

F\left(x_{1}, \ldots , x_{n}\right) \geqslant \prod _{k = 1}^{n} F_{k}\left(x_{k}\right)

for any $x_{1}, \ldots , x_{n}$ .

?

Problem 3.339

Prove that for no positive integer $n \geqslant 2$ , on no probability space, do there exist three positive integer-valued random variables $\xi , \eta , \zeta$ , each of which assigns positive probability to every number $1,2, \ldots$ , such that $\xi$ and $\eta$ are independent and

\xi ^{n}+\eta ^{n} = \zeta ^{n}

?

Problem 3.340

Is the following reasoning valid: "From home to work is 1 km, I walk on average at a speed of $5 \text{ km/h}$ , hence on average my commute will take 12 minutes"?

?