Math Problems Vault

Problem 10.1

The two-dimensional distribution of a pair of integer-valued random variables $\xi$ and $\eta$ is given by the table

\begin{array}{c|ccc} {}_\eta \backslash ^\xi & -1 & 0 & 1 \\ \hline -1 & 1/8 & 1/12 & 7/24 \\ 1 & 5/24 & 1/6 & 1/8 \end{array}

where the entry at the intersection of column $\xi = i$ and row $\eta = j$ is the probability $\mathbb {P}\left\{ \xi = i, \eta = j\right\}$ . Determine whether the random variables $\xi$ and $\eta$ are dependent. Find:

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(a)

the one-dimensional distributions $\mathbb {P}\left\{ \xi = i\right\}$ and $\mathbb {P}\left\{ \eta = j\right\}$ ;

(b)

the conditional probabilities $\mathbb {P}\left\{ \eta = 1 \mid \xi = 0\right\}$ and $\mathbb {P}\left\{ \xi = 1 \mid \eta = 1\right\}$ ;

(c)

the joint distribution of the quantities $\xi + \eta$ and $\xi \eta$ ;

(d)

the one-dimensional distributions of the quantities $\xi + \eta$ and $\xi \eta$ ;

(e)

the joint distribution of the quantities $\max (\xi , \eta )$ and $\min (\xi , \eta )$ ;

(f)

the joint distribution of the quantities $\left|\xi + 2\eta \right|$ and $\left|\xi - \eta \right|$ .

Problem 10.2

Solve problem 10.1 for the table

\begin{array}{c|ccc} {}_\eta \backslash ^\xi & -2 & 0 & 1 \\ \hline 1 & 1/6 & 1/6 & 1/6 \\ 2 & 1/6 & 1/6 & 1/6 \end{array}

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(a)

(b)

(c)

(d)

(e)

Problem 10.3

Solve problem 10.1 for the table

\begin{array}{c|ccc} {}_\eta \backslash ^\xi & -1 & 0 & 1 \\ \hline -2 & 1/8 & 1/12 & 7/24 \\ 0 & 1/12 & 1/12 & 1/16 \\ 1 & 1/8 & 1/12 & 1/16 \end{array}

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(a)

(b)

(c)

(d)

(e)

(f)

Problem 10.4

(a)(b)(c)

A number $\xi$ is chosen at random among the numbers 1, 2, and 3; a number $\eta$ is chosen at random among $\xi , \xi +1, \ldots , \xi +3$ . Find:

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(a)

the joint distribution of the numbers $\xi$ and $\eta$ ;

(b)

the one-dimensional distribution of $\xi$ ;

(c)

the one-dimensional distribution of $\eta$ .

Problem 10.5

Let the random variables $\xi$ and $\eta$ be independent and identically distributed, with $\mathbb {P}\left\{ \xi = 1\right\} = p$ and $\mathbb {P}\left\{ \xi = 0\right\} = 1 - p$ . Introduce a new random variable $\zeta$ , setting

\zeta = \begin{cases} 0, & \text{if } \xi + \eta \text{ is even}, \\ 1, & \text{if } \xi + \eta \text{ is odd.} \end{cases}

For which values of $p$ are the random variables $\xi$ and $\zeta$ independent?

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Problem 10.6

Let $\xi_1$ and $\xi_2$ be independent random variables taking integer values. Under what conditions are the events $\left\{ \xi_1 \text{ is even}\right\}$ and $\left\{ \xi_1 + \xi_2 \text{ is even}\right\}$ independent?

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Problem 10.7

(a)(b)(c)

Let the independent random variables $\xi$ , $\eta$ , and $\zeta$ have the same geometric distribution with parameter $p$ . Find:

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(a)

$\mathbb {P}\left\{ \xi = \eta \right\}$ ;

(b)

$\mathbb {P}\left\{ \xi \geqslant 2\eta \right\}$ ;

(c)

$\mathbb {P}\left\{ \xi + \eta \leqslant \zeta \right\}$ .

Problem 10.8

(a)(b)(c)

A point $\omega = (\omega_1, \omega_2)$ is chosen at random in the square $[0, 1]^2$ . Find the distribution function and density of the random variable $\xi + \eta$ if:

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(a)

$\xi (\omega ) = \omega_1 + \omega_2$ , $\eta (\omega ) = \omega_1 - \omega_2$ ;

(b)

$\xi (\omega ) = \omega_1$ , $\eta (\omega ) = \omega_2$ ;

(c)

$\xi (\omega ) = 1$ if $\omega_1 = \omega_2$ and $\xi (\omega ) = 0$ if $\omega_1 \neq \omega_2$ ; $\eta (\omega ) = \omega_1\omega_2$ .

Problem 10.9

(a)(b)

Consider in the plane a region $\Omega$ of area $1/2$ that is the union of the quadrilateral with vertices $(0,0)$ , $(1,1)$ , $(0,1/2)$ , $(1/2,1)$ and the triangle with vertices $(1/2,0)$ , $(1,0)$ , $(1,1/2)$ . A random point $(\xi , \eta )$ is uniformly distributed in $\Omega$ . Prove that:

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(a)

$\xi$ and $\eta$ are uniformly distributed on $[0, 1]$ ;

(b)

the sum $\xi + \eta$ has the same density as it would if $\xi$ and $\eta$ were independent.

Problem 10.10

(a)(b)(c)

For a given tram line, the function $F(a, b)$ is known, equal to the probability that a passenger riding this line boarded at a point $x < a$ and rides to a point $y \leqslant b$ . Find the probability that a passenger:

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(a)

boarded the tram before point $z$ ;

(b)

got off before point $z$ ;

(c)

passes through point $z$ .

Problem 10.11

Let $F(x, y)$ denote a two-dimensional distribution function. Set $G(x, y) = 0$ for $x < 0$ and $y < 0$ ; $G(x, y) = F(x, y)$ at all other points. Prove that $G$ is monotone in each variable, but $G$ need not be a distribution function.

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Problem 10.12

Let $\xi$ and $\eta$ be random variables with distribution functions $F$ and $G$ respectively. Prove that if $\mathbb {P}\left\{ \xi \leqslant \eta \right\} = 1$ , then $F(x) \geqslant G(x)$ for all $x$ .

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Problem 10.13

Two points are thrown independently at random onto the segment $[0, a]$ . Find the distribution function of the distance between them.

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Problem 10.14

Let $p(x, y)$ be the joint density of the random variables $\xi$ and $\eta$ . Find:

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(a)

$\mathbb {P}\left\{ \xi > \eta \right\}$ ;

(b)

$\mathbb {P}\left\{ \xi > \left|\eta \right|\right\}$ ;

(c)

$\mathbb {P}\left\{ \max (\xi , \eta ) < a\right\}$ ;

(d)

$\mathbb {P}\left\{ \left|\xi \right| > \eta \right\}$ ;

(e)

$\mathbb {P}\left\{ \xi - \eta > 1\right\}$ ;

(f)

$\mathbb {P}\left\{ \min (\xi , \eta ) < a\right\}$ .

Problem 10.15

(a)(b)

Let the random variables $\xi$ and $\eta$ be independent and have the same continuous distribution function. Find:

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(a)

$\mathbb {P}\left\{ \xi = \eta \right\}$ ;

(b)

$\mathbb {P}\left\{ \xi < \eta \right\}$ .

Problem 10.16

Let the independent random variables $\xi$ and $\eta$ have exponential distributions with parameters $\alpha$ and $\beta$ respectively. Find $\mathbb {P}\left\{ \xi < \eta \right\}$ .

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Problem 10.17

The joint distribution of the random variables $\xi$ and $\eta$ has density

p(x, y) = \begin{cases} c(x+y) & \text{for } 0 \leqslant x \leqslant 1,\ 0 \leqslant y \leqslant 1, \\ 0 & \text{otherwise.} \end{cases}

Find:

?

(a)

the constant $c$ ;

(b)

the density of $\xi$ ;

(c)

the density of $\eta$ ;

(d)

the density of $\max (\xi , \eta )$ ;

(e)

the density of $\min (\xi , \eta )$ .

Problem 10.18

The random variable $\xi$ has probability density $p(x)$ . Find the joint distribution function of the random vector $(\xi , \xi^2)$ .

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Problem 10.19

The joint distribution of the random variables $\xi$ and $\eta$ has density

p(x, y) = \begin{cases} 1, & \text{if } x, y \geqslant 0 \text{ and } x + 2y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}

Find the density of the random variable $\xi$ .

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Problem 10.20

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

The joint distribution of the random variables $\xi$ and $\eta$ has density

p(x, y) = \frac{c}{1 + x^2 + x^2y^2 + y^2}.

Determine whether the random variables $\xi$ and $\eta$ are dependent or not. Find:

?

(a)

the constant $c$ ;

(b)

the density of $\xi$ ;

(c)

the density of $\eta$ ;

(d)

the probability $\mathbb {P}\left\{ \left|\xi \right| \leqslant 1, \left|\eta \right| \leqslant 1\right\}$ .

Problem 10.21

(a)(b)(c)

A random point $(\xi , \eta )$ is uniformly distributed inside the square $\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant 1\right\}$ . Determine whether the random variables $\xi$ and $\eta$ are dependent or not. Find:

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(a)

the density of $\xi$ ;

(b)

the density of $\eta$ ;

(c)

the probability $\mathbb {P}\left\{ \xi > \eta \right\}$ .

Problem 10.22

The joint distribution of the random variables $\xi$ and $\eta$ has density

p(x, y) = \begin{cases} \dfrac {2}{\pi (x^2+y^2)^3} & \text{for } x^2 + y^2 \geqslant 1, \\ 0 & \text{otherwise.} \end{cases}

Find the density of $\sqrt{\xi^2 + \eta^2}$ .

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Problem 10.23

A random point $(\xi , \eta )$ has a normal distribution on the plane with density

p_{\xi \eta }(x, y) = \frac{1}{2\pi \sqrt{2}} e^{-(2(x-1)^2 + (y+1)^2)/4}.

Find the probability that the random point $(\xi , \eta )$ lands in the region bounded by the ellipse

\frac{(x-1)^2 + (y+1)^2}{2} = 1.

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Problem 10.24

(a)(b)

The random variables $\xi$ and $\eta$ are independent and have the standard normal distribution. Find the probability that the random point $(\xi , \eta )$ lands

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(a)

in the annulus $\left\{ (x, y) : 2 \leqslant \sqrt{x^2+y^2} \leqslant 3\right\}$ ;

(b)

in the square $\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant \sqrt{2}\right\}$ .

Problem 10.25

(a)(b)

A random point $(\xi , \eta )$ has a uniform distribution in the disk $\left\{ (x, y) : x^2 + y^2 < 1\right\}$ .

?

(a)

Find the probability $\mathbb {P}\left\{ \left|\xi \right| < 1/\sqrt{2}, \left|\eta \right| < 1/\sqrt{2}\right\}$ .

(b)

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

Problem 10.26

The random vector $(\xi , \eta , \zeta )$ has probability density

p(x, y, z) = \begin{cases} \dfrac {6}{(1+x+y+z)^4} & \text{for } x > 0,\ y > 0,\ z > 0, \\ 0 & \text{otherwise.} \end{cases}

Find the distribution of the quantity $\xi + \eta + \zeta$ .

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Problem 10.27

Two points are chosen randomly and independently on the segment $[0, 1]$ ; a third point is chosen at random between the first two. Prove that the distribution of the coordinate of the third point is absolutely continuous. Find its density.

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Problem 10.28

Show that if $\xi_1, \xi_2, \ldots$ are independent random variables, then both random variables $\liminf_{n \to \infty } \xi_n$ and $\limsup_{n \to \infty } \xi_n$ are degenerate.

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Problem 10.29

Let the random variables $\xi$ and $\eta$ be such that $\mathbb {P}\left\{ \xi > 0\right\} = \mathbb {P}\left\{ \eta > 0\right\} = 3/4$ and $\mathbb {P}\left\{ \xi + \eta > 0\right\} = 1/2$ . Prove that $\xi$ and $\eta$ are dependent.

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Problem 10.30

Let $\xi$ have the standard normal distribution, and let $\eta = 1$ if $\xi > 0$ and $\eta = 0$ if $\xi \leqslant 0$ . Prove that the random variables $\left|\xi \right|$ and $\eta$ are independent.

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Problem 10.31

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

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

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(a)

$\max (\xi , \eta )$ ;

(b)

$\min (\xi , \eta )$ ;

(c)

$\max (3\xi , \eta )$ ;

(d)

$\min (\xi , \eta^3)$ .

Problem 10.32

(a)(b)(c)

Three customers approach a telephone exchange with two booths: the first and second customers occupy booths No.1 and 2 respectively, and the third customer is left waiting. Assuming the call durations $\tau_1$ , $\tau_2$ , and $\tau_3$ of the customers are independent and exponentially distributed with parameter $\alpha$ , find:

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(a)

the probability that the third customer will get booth No.1;

(b)

the density of the waiting time of the third customer;

(c)

the probability that the third customer finishes talking before the first or the second customer (i.e., before at least one of them).

Problem 10.33

When is a random variable $\xi$ independent of itself?

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Problem 10.34

Under what conditions on $\xi$ are the random variables $\xi$ and $\sin \xi$ independent?

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Problem 10.35

Prove that if the random variables $\xi$ and $f(\xi )$ ( $f$ a Borel function) are independent, then $f(\xi )$ equals some constant with probability 1.

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Problem 10.36

Prove that if the random variables $\xi$ and $\eta$ , and also $\xi$ and $\xi - \eta$ , are independent, then $\xi$ equals some constant with probability 1.

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Problem 10.37

Let $\xi$ and $\eta$ be random variables. Must they be independent if the random variables $\xi^2$ and $\eta^2$ are independent?

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Problem 10.38

Let the independent random variables $\xi$ , $\eta$ , and $\zeta$ take only the values $0, 1, 2, \ldots$ , each with positive probability. Prove that there is no integer $n \geqslant 2$ for which the equality $\xi^n + \eta^n = \zeta^n$ holds with probability 1.

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Problem 10.39

Let $\xi$ and $\eta$ be independent random variables, with $\xi + \eta$ taking the values 0, 1, and 2 each with probability $1/3$ . Prove that one of the variables $\xi$ or $\eta$ has a degenerate distribution.

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