Complements and Problems

The density of the impact point $(X, Y)$ of a circular target is given by

Find, (a) the constant $c$, (b) the probability $p$ that the impact point falls in a circle with radius $a < R$ centered at the origin.

The random variables $X, Y$ satisfy the linear relation

where $a, b, c$ are constants. Find, (a) the correlation $\rho (X, Y)$, and (b) the ratio $\sigma_{x}/\sigma_{y}$ of the standard deviations.

The dispersion (variance--covariance) matrix of a three-dimensional normal distribution is

If $\mu_{x} = \mu_{y} = \mu_{z} = 0$, find the density $f(x, y, z)$ and its maximum.

The joint density of $X$ and $Y$ is given by

Find, (a) the marginal densities of $X$ and $Y$, and (b) the conditional distribution function $F_{X}(x \mid y)$ of $X$ given $Y = y$.

Suppose that the random variable $X$ coincides with the random variable $X_{i}$ with probability $p_{i}$, that is,

Show that

where $\mathbb {P}\left(M = m_{i}\right) = p_{i}$ $(i = 1, 2, \ldots )$.

The random vector $(X, Y, Z)$ has density $f(x, y, z)$. $X, Y, Z$ can only be observed simultaneously. An observation gave the values $u, v, w$ without knowing the correspondence between them and $X, Y, Z$. Find the probability that $X = u$, $Y = v$, and $Z = w$.

The joint distribution function $F(x, y)$ of the random variables $X$ and $Y$ is given by

Find, (a) the density function $f(x, y)$, and (b) the dispersion matrix.

The normal density of a point $A = (X_{1}, X_{2}, X_{3})$ is given by

What is the probability that $A$ will fall into an ellipsoid, with principal semiaxes $E_{1}, E_{2}, E_{3}$ along the axes $Ox, Oy, Oz$, where $E_{1}, E_{2}, E_{3}$ are the probable errors of $X, Y, Z$, respectively, i.e.,

The vector $(X, Y)$ has the uniform distribution in the ellipse

Find:

The point $(X, Y, Z)$ is uniformly distributed in the sphere $x^{2} + y^{2} + z^{2} \leq R^{2}$. Find the density of $Z$ and the conditional density $f(x, y \mid z)$. Generalize for an $n$-dimensional sphere.

Given $f_{Y}(y)$, $\mathbb {E}\left[ X \mid y \right]$, $\operatorname {Var}\left[ X \mid y \right]$, find $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.

The vector $(X, Y)$ has normal density

Find the densities $f_{X}(x)$, $f_{Y}(y)$, $f_{X}(x \mid y)$, $f_{Y}(y \mid x)$ and the constant $c$.

Suppose that $\overrightarrow {X}$ has a spherically symmetric $n$-dimensional distribution, i.e., with density $f(x_{1}^{2} + \cdots + x_{n}^{2})$. Prove that the density $g$ of the ``generalized $\chi^{2}$ variable'' $U = \overrightarrow {X}'\overrightarrow {X}$ is given by

where $C_{n}$ is the ``surface'' area of the $n$-dimensional sphere, i.e.,

(Continuation of Problem 1.13.) For $n = 2$, show that the polar coordinates $(R, \theta )$, where

are independent, $\theta$ is uniformly distributed on $[0, 2\pi )$, and $R$ has density (cf. Exercise 221)

(Continuation of Problem 1.14.) Show that the correlation coefficient based on the single observation $(X, Y)$, i.e.,

has density

(Continuation of Problem 1.15.) If $\overrightarrow {X} = (X_{1}, \ldots , X_{n})'$ is $\mathscr {N}\left(0, I_{n}\right)$ and $\overrightarrow {X}$ is partitioned into $k$ subvectors $X_{(1)}, \ldots , X_{(k)}$ with $v_{1}, v_{2}, \ldots , v_{k}$ components, respectively, show that $(X_{(1)}'X_{(1)}, \ldots , X_{(k)}'X_{(k)})/\left|X\right|^{2}$ has the Dirichlet distribution (see (5.11)), with density $f(t_{1}, \ldots , t_{k-1})$ given below.

Formula (5.11), Dirichlet distribution: for $x_{1}, \ldots , x_{k}$ with $\sum_{i=1}^{k+1} x_{i} = 1$, $x_{i} \geq 0$,

The density $f(t_{1}, \ldots , t_{k-1})$ asked for here is

Hence, for $v_{1} = v_{2} = \cdots = v_{s} = 1$, $v_{s+1} = n - s$ $(k = s+1)$,

has the density

and the density of $(X_{1}, \ldots , X_{k})/\left|X\right| \equiv (U_{1}, U_{2}, \ldots , U_{k})$ is

This is the distribution of any $k$-dimensional subvector of an $n$-dimensional vector

uniformly distributed on the (surface of the) unit sphere

(Continuation of Problem 1.16.) In general, if $\overrightarrow {X}$ has a spherical distribution (see Problem 1.13), since this is invariant under rotations in $n$-space, the distribution of the direction $\overrightarrow {X}/\left|X\right|$ is the same as that of $\overrightarrow {U}$ and is independent of its length $\left|X\right|$ (for any spherical distribution with $\mathbb {P}\left(\left|X\right| = 0\right) = 0$).

Hence conclude that the results of Problem 1.16 hold if $\overrightarrow {X}$ is any spherically symmetric random vector.

Let $X, Y$ be random variables with

Find an approximate formula for

using the linear expansion of $g(x, y)$ around $(\mu , \eta )$ (up to second-order moments). Generalize for $n$ random variables.

$X$ and $Y$ are independent and identically distributed random variables with density given by

Using Problem 1.18 find the $\mathbb {E}\left[Z\right]$ and $\operatorname {Var}\left[Z\right]$ where

The fundamental frequency of a chord is given by the formula

where $F$ is the tension, $m$ is the mass of the chord, and $L$ is its length. If the mass is assumed to be constant and $F, L$ are random variables with $\mathbb {E}\left[F\right] = f$, $\mathbb {E}\left[L\right] = l$, $\operatorname {Var}\left[F\right] = \sigma_{f}^{2}$, $\operatorname {Var}\left[L\right] = \sigma_{l}^{2}$, and $\operatorname {Cov}\left[ F, L \right] = \sigma_{fl}$, find an approximation for the variance of $\Omega$.

Let the vector $(X, Y)$ have density $f(x, y)$. We define the complex variables

and $Z_{t} = Ze^{it}$. Prove that, in order for all $Z_{t}$ to have the same distribution, it is necessary that $f(x, y) = g(x^{2} + y^{2})$ for some density $g$ (see Problem 1.13).

If $\overrightarrow {X}$ has density $f$ as in Problem 1.13, show that

has density of the form

and characteristic function of the form

Let $\rho_{ij} = \rho (X_{i}, X_{j})$ be the correlation coefficient between $X_{i}, X_{j}$ for $i, j = 1, \ldots , n$. For $n = 3$ and $\rho_{12} = \rho_{13} = \rho_{23} = \rho$, what are the possible values that $\rho$ can take?

If the $p$ vector $\overrightarrow {X}$ is $\mathscr {N}\left(\mu , \Lambda \right)$, show that the quadratic form

has the $\chi_{p}^{2}$ distribution, with $p$ degrees of freedom, and density given by (cf. Problem 1.13)

The entropy $H$ of a multivariate distribution with density $f(\overrightarrow {x})$ is defined by the formula

Prove that for the normal distribution of Problem 1.24

If $\overrightarrow {X}$ is $\mathscr {N}\left(\mu , \Lambda \right)$ with density $f$, find the density $g$ of the random variable

If $\overrightarrow {X} = (X_{1}, \ldots , X_{n})$ is uniformly distributed in the simplex $T$:

show that the density of $\overrightarrow {X}$ is

Show that the $(n-1)$-dimensional marginals are also special cases of the Dirichlet distribution (5.11).

Formula (5.11), Dirichlet distribution: for $x_{1}, \ldots , x_{k}$ with $\sum_{i=1}^{k+1} x_{i} = 1$, $x_{i} \geq 0$,

Two persons shoot at a target, each has $n$ shots. Using generating functions, find the probability $p$ that they have the same number of successes, given that each of them has a probability $0.5$ of success (hitting the target) at each shot.

To obtain the title of champion in chess, an opponent has to win at least $12.5$ points out of the maximum of $24$. In the case of a tie ($12:12$), the defending champion maintains the title. The probability that each of the two players wins a game is the same, and equals half the probability of a tie. Find:

In a lottery, tickets are numbered from $000000$ to $999999$ and they are all equally probable. What is the probability $p$ that the sum of the digits of a chosen number be $21$?

Prove that the Pascal distribution, for $r \to \infty$ and $rq \to \lambda$, converges to the Poisson with parameter $\lambda$.

Let $X, Y, Z$ be independent random variables, each taking the values $1, 2, \ldots , n$ with the same probability (uniformly). Find $\mathbb {P}\left(X + Y = 2Z\right)$ from their joint generating function $G(t_{1}, t_{2}, t_{3})$.

In Exercise 315, let

Find the generating function $Q(t)$ of $q_{n,k}$ for a given $n$ and hence $q_{n,k}$.

(Continuation of Problem 2.6.) Suppose $k$ and $m$ tend to infinity so that $k/m \to x$. Find the limit of $q_{n,k}$.

Find the densities of the distributions with characteristic functions

Find the moments of the Laplace distribution with characteristic function

Determine the distribution with characteristic function

Find the characteristic function of $Y = aF(X) + b$, where $X$ is a continuous random variable and $F$ is its distribution function.

A vector $\xi = (X_{1}, \ldots , X_{n})$ follows the $n$-dimensional normal distribution with $\mathbb {E}\left[X_{i}\right] = a$, $\operatorname {Var}\left[X_{i}\right] = \sigma^{2}$ $(i = 1, \ldots , n)$ and covariances

Find the characteristic function of $\xi$.

Let $\varphi (t)$ be the characteristic function of a random variable $X$ with $\mathbb {E}\left[X^{2}\right] = \mu_{2}'$.

Show that

is the characteristic function of a random variable $X$, and determine the corresponding distribution.

Let $X_{1}, X_{2}, X_{3}$ be independent $\mathscr {N}\left(0, 1\right)$. Find the characteristic function of the pair $Z_{1}, Z_{2}$, where

Hence conclude the characteristic function of $Z_{1}$. Are $Z_{1}, Z_{2}$ independent?

Find the distributions corresponding to the characteristic functions

Show that

with positive parameters and $p_{0} + p_{1} + p_{2} = 1$ is the probability generating function of a pair $(X, Y)$, so that the marginal distributions of $X$ and $Y$ and $X + Y$ are negative binomials.

Let a ``round'' consist of a sequence of Bernoulli trials before the first failure. Find the generating function and the probability function of the total number $S_{k}$ of successes in $k$ rounds.

(Continuation of Problem 2.18.) Let $G$ be the number of successive rounds up to the $n$th success (i.e., the $n$th success occurs during the $G$th round). Find the distribution of $G$ and the

(Continuation of Problem 2.19.) Consider two independent sequences of Bernoulli variables with parameters $p_{1}$ and $p_{2}$, respectively, or two players with ``skills'' $p_{1}$ and $p_{2}$, respectively. Prove that each player will need the same number of rounds (trials) until the $N$th success (a tie) with probability

Show that the characteristic function of the $\beta$ density (2.10) is

Formula (2.10), the beta density on $(0,1)$ with parameters $p, q > 0$:

Show that the characteristic function of the density in Problem 1.13 is of the form $\varphi (t_{1}^{2} + t_{2}^{2} + \cdots + t_{n}^{2})$ (see Problem 1.21). If

that is, $X_{1}, X_{2}, \ldots , X_{n}$ are independent, prove that $\overrightarrow {X}$ is $\mathscr {N}\left(0, I\right)$. In other words, a spherically symmetric random vector has independent components if and only if it is multivariate normal.

Find the density of

where $X \sim N(0, 1)$.

$X$ has a uniform distribution on $(0, 1)$. Find the density of the random variable $Y$ defined by

The random variables $X$ and $Y$ are related by

where $f(t) \geq 0$ satisfies

Prove that if $X$ is uniform on $(0, 1)$, then $f(t)$ is the density of $Y$ (see Exercise 227).

Suppose $X$ and $Y$ are independent random variables. Prove that the product $XY$ has the same distribution if:

The roots of $x^{2} + Yx + Z = 0$ follow the normal distribution $(-1, 1)$. Find the density of the coefficients $Y$ and $Z$ (cf. Exercise 222).

Let $f(x, y, z)$ be the density of a point $(X, Y, Z)$ in the three-dimensional space and

What is the density of the pair $(R, \theta )$?

$X$ and $Y$ are independent normal $N(0, \sigma^{2})$. Find the distribution of

$X$ and $Y$ are independent random variables with uniform distribution on $(0, a)$. Find the distribution of $Z = X/Y$ and examine the existence of moments of $Z$.

The joint density of $X, Y$ is

$X, Y$ are independent exponentials each with density

If the random variables $X_{i}$ are independent normal $N(a + bt_{i}, \sigma^{2})$ $(i = 1, \ldots , n)$, where $t_{i}$ are constants with $\sum_{i=1}^{n} t_{i} = 0$, find:

(Continuation of Problem 3.11.) By a proper orthogonal transformation of $\mathbf{X} = (X_{1}, \ldots , X_{n})'$, say $\mathbf{Y} = 0\mathbf{X}$ ($0$ an $n \times n$ orthogonal matrix), prove that the random variables $\overline{X}$, $b$ and $\sum_{i=1}^{n} (X_{i} - \overline{X} - bt_{i})^{2}$ are completely independent. Conclude that $\sum_{i=1}^{n} (X_{i} - \overline{X} - bt_{i})^{2}/\sigma^{2}$ has the $\chi^{2}$ distribution with $n - 2$ degrees of freedom.

$X$ and $Y$ are jointly normal with $\mathbb {E}\left[X\right] = \mathbb {E}\left[Y\right] = 0$, $\operatorname {Var}\left[X\right] = \operatorname {Var}\left[Y\right] = 1$, and $\operatorname {Corr}\left[ X, Y \right] = \rho$, $\left|\rho \right| < 1$. Consider the polar coordinates $R = \sqrt{X^{2} + Y^{2}}$ and $\theta = \tan^{-1}(Y/X)$. Show that $\theta$ has density (cf. Exercises 220, 221)

The random vector $\mathbf{X} = (X_{1}, \ldots , X_{k})$ has the density

If $\Lambda = I$ and $k = 1$ this is the Student distribution with $n$ degrees of freedom. If $\Lambda = I_{k}$, then $X$ has a spherically symmetric $t$ distribution (see Problem 1.13). Show that:

Let $f(x)$ denote the density of a random variable $X$. If $f(\cdot )$ is monotone and bounded, then $f(X)$ has a uniform distribution if and only if $X$ has the (negative) exponential distribution.

A Geiger--Muller counter registers a particle emitted by a radioactive source with probability $10^{-4}$.