Multivariate Distributions

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 $\sum_{i=1}^{k+1} x_{i} = 1$, $x_{i} \geq 0$,