Advanced Topics

The joint distribution of $(X, Y)$ is defined by $\mathbb {P}\left(X=0, Y=0\right) = \mathbb {P}\left(X=0, Y=1\right) = \mathbb {P}\left(X=1, Y=1\right) = 1/3$.

Show that the function

is not a joint distribution function.

We consider a family with two children. Let $X_{n} = 1$ if the $n$th child is a boy for $n = 1, 2$ and $X_{3} = 1$ if there is only one boy, otherwise $X_{i} = 0$ ($i = 1, 2, 3$). Show that the $X_{i}$ are pairwise independent but not completely independent.

$X$ and $Y$ have the joint density

Find

The $k$-dimensional hypergeometric distribution has probability function

where

Show that the conditional distribution of $X_{k}$ given $X_{1}, \ldots , X_{k-1}$ is hypergeometric.

If $X_{1}, X_{2}, \ldots , X_{k}$ are distributed according to the multinomial distribution, the conditional distribution of $X_{1}$, given $X_{2}=n_{2}, \ldots , X_{k}=n_{k}$, is binomial with parameters $n-(n_{2}+\cdots +n_{k})$ and $p_{1}/(p_{1}+p_{k+1})$.

If $X_{1}, X_{2}, \ldots , X_{k}$ have a Dirichlet distribution, show that the conditional distribution of $X_{k}/S_{k}$, for $X_{i}=x_{i}$, where $S_{k} = 1-(X_{1}+\cdots +X_{k-1})$ is $\beta (n_{k}, n_{k+1})$.

Let $X = (X_{1}, X_{2}, X_{3})$ be uniformly distributed in the subset of the positive orthant defined by $x_{1} \geq 0$, $x_{2} \geq 0$, $x_{3} \geq 0$, $x_{1}+x_{2}+x_{3} \leq c$. Find,

A bivariate normal distribution has density

Recall that the $n$-dimensional (nonsingular) normal distribution $N(\mu , \Sigma )$ has density

where $\mu = (\mu_{1}, \ldots , \mu_{n})' = \mathbb {E}\left[X\right]$ denotes the mean vector and $\Sigma$ denotes the (positive definite) covariance matrix of the normal random vector $X$.

If $(X, Y)$ is uniform in the triangle $x \geq 0$, $y \geq 0$, $x+y \leq 2$, find,

A die is thrown 12 times.

Show that the most probable value $(n_{1}^{0}, n_{2}^{0}, \ldots , n_{k+1}^{0})$ of the multinomial distribution satisfies the relations

If the number of trials $n$ is large and the $p_{i}$ small so that the $np_{i} = \lambda_{i}$ are moderate, show that the multinomial distribution can be approximated by the so-called multiple Poisson distribution

Given that the density of $X$ and $Y$ is

find,

Find the density function $f(x, y)$ of the uniform distribution in the circle $x^{2}+y^{2} \leq 1$. Find the marginal distributions of $X$ and $Y$. Are the variables $X$ and $Y$ independent?

The joint density of the variables $X, Y, Z$ is

Find $\mathbb {P}\left(X < Y < Z\right)$.

For each of the following densities $f(x, y)$, find $F(x, y)$, $F_{X}(x)$, $F_{Y}(y)$, $f_{X}(x)$, $f_{Y}(y)$, $f_{X}(x \mid Y=y)$, $f_{Y}(y \mid X=x)$.

The joint probability function $p(x_{i}, y_{j})$ of $X, Y$ is given in the following table:

Show that the discrete variables $X$ and $Y$ with joint probability function $p_{ij} = \mathbb {P}\left(X=x_{i}, Y=y_{j}\right)$ are independent if and only if the matrix of probabilities $P = (p_{ij})$ has rank 1.

Show that two orthogonal (uncorrelated) variables each taking two values are independent.

Find the probability generating function and hence the mean and variance of the following distributions: (a) binomial, (b) Poisson, (c) geometric, (d) Pascal. Moreover, deduce the corresponding moment generating functions and characteristic functions.

Let $X_{1} = X_{2} = X$ when $X$ has the Cauchy density

Prove that

while $X_{1}$, $X_{2}$ are not independent.

From the probability generating function $P_{X}(t)$ of $X = 0, 1, 2, \ldots$, find the probability generating function of $Y = 3X+2$.

Using the probability generating function prove that the random variable $S_{N}$ of formula (6.3) --- $S_{N} = X_{1}+X_{2}+\cdots +X_{N}$, where the random variable $N = 0, 1, \ldots$ and the $X_{i}$ are completely independent and identically distributed --- has variance given by $\operatorname {Var}\left[S_{N}\right] = \mathbb {E}\left[N\right]\operatorname {Var}\left[X\right] + \operatorname {Var}\left[N\right]\mathbb {E}\left[X\right]^{2}$.

A woman continues to have children until she has a boy! Suppose that the probability of having a blond child is $p$; find the probability that the woman will have $k$ blond children. (Assume that the color is independent of sex.)

The number $N$ of visitors to a shop has the Poisson distribution. On average, 150 women visit the shop every day. Of these, $100p_{1}\%$ buy mini, $100p_{2}\%$ buy midi, $100p_{3}\%$ buy maxi, and the remaining buy nothing. (No woman buys more than one dress!!) Let $X_{1}, X_{2}, X_{3}$ be the corresponding numbers of mini, midi, and maxi that are bought in a day. Find the joint probability function of $X_{1}, X_{2}, X_{3}$. Are they independent?

Let $X$, $Y$ be independent with densities

Prove that $\varphi_{X}(t) = \varphi_{Y}(t)$ for $|t| \leq \min \left\{ 1/a, 1/b, 1/c\right\}$, $2c = a+b$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent random variables having the same normal distribution with mean $\mu$ and variance $1$. We set

Moreover, let the random variable $T = aY_{v}$ where $a, v$ are suitable constants, and $Y_{v}$ follows the $\chi^{2}$ distribution with $v$ degrees of freedom. Using characteristic functions, find the constants $a$ and $v$ so that the random variables $S_{n}$ and $T$ have the same means and the same variances.

Prove that $S_{N}$ of formula (6.3) --- $S_{N} = X_{1}+X_{2}+\cdots +X_{N}$, where the random variable $N = 0, 1, \ldots$ and the $X_{i}$ are completely independent and identically distributed --- has the characteristic function

Prove that $\varphi (t) = \exp [\lambda (e^{-|t|}-1)]$ is the characteristic function of a Poisson distribution compounded with (generalized by) a Cauchy distribution. Then, using the inversion formula, find its density.

Let $Z_{n}$ be a random variable following the binomial distribution with parameters $n$ and $p = \lambda /n$ $(\lambda > 0)$. Moreover, let the random variable $Z$ have a Poisson distribution with parameter $\lambda$. Prove that $\lim_{n\to \infty } \varphi_{Z_{n}}(t) = \varphi_{Z}(t)$ $(t \in R)$, where $\varphi_{Z_{n}}(t)$ and $\varphi_{Z}(t)$ are the characteristic functions of $Z_{n}$ and $Z$, respectively. Then deduce the approximation of the binomial by the Poisson.

Calculate the characteristic function of a Gamma distribution with density

Then deduce the characteristic function of $\chi_{n}^{2}$.

At the $n$th toss of a coin a player receives or pays $2^{-n}$ of a drachma depending on whether heads or tails appears. Let $Y_{n}$ be the gain of the player after $n$ tosses. Show that the distribution of $Y_{n}$ as $n\to \infty$ tends to the uniform distribution in the interval $(-1,1)$. First find the characteristic function of $Y_{n}$.

$X$ has distribution function $F(x)$ which is a mixture of two distribution functions $F_{1}, F_{2}$ as follows:

If $F_{1}$ is $N(\mu_{1}, \sigma_{1}^{2})$ and $F_{2}$ is $N(\mu_{2}, \sigma_{2}^{2})$ find the characteristic function of $F$ and then the $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.

Find the second characteristic function (cumulant generating function) $\psi (t)$ of the exponential with density

and prove that the cumulant of order $r$, $\kappa_{r}$, is given by

(Continuation of Problem 205.) Let $Y$ be a discrete random variable defined as follows. For $h > 0$ and $k = 0, 1, 2, \ldots$,

Find the probability generating function of $Y$ and then show that $\mathbb {E}\left[X\right] > \mathbb {E}\left[Y\right]$, $\operatorname {Var}\left[X\right] < \operatorname {Var}\left[Y\right]$.

Suppose that the density $f(x,y) = g(x^{2}+y^{2})$, i.e., it has circular symmetry about the origin. Show that the characteristic function of $f$, $\varphi (t,u)$, is a function of $t^{2}+u^{2}$ (only). Hence show that the only distribution of independent random variables with the above property is the normal. (Consider an orthogonal transformation of $t, u$.)

Consider two independent random samples $X_{1}, X_{2}, \ldots , X_{n}$ and $Y_{1}, Y_{2}, \ldots , Y_{n}$ from the Laplace distribution

Show that the means of the differences $X_{i}-Y_{i}$ and $X_{i}+Y_{i}$ have the same distribution but are not independent. Are they orthogonal?

If $\varphi (t)$ is a characteristic function show that, (a) $|\varphi (t)|^{2}$, (b) $e^{\lambda (\varphi (t)-1)}$ are also characteristic functions.

Calculate the probability generating function of the multinomial distribution and then show that

(Continuation of Problem 210.) If, as $n\to \infty$, $np_{j}\to \lambda_{j}$ $(j=1,2,\ldots ,k+1)$, then the multinomial distribution tends to the $k$-dimensional Poisson $Y=(Y_{1},Y_{2},\ldots ,Y_{k})$ where the $Y_{i}$ are independent Poisson, $Y_{j}$ with parameter $\lambda_{j}$ (cf. Exercise 183).

Calculate the characteristic function of the negative multinomial distribution

where $s>0$, $\theta_{j}>0$ $(n_{j}=0,1,2,\ldots )$, and $n_{k+1}=n-(n_{1}+\cdots +n_{k})$, and hence

$X, Y$ have a distribution of the continuous type with characteristic function $\varphi (t_{1},t_{2})$. Show that the conditional characteristic function of $X$ given $Y=y$ is

A random sample $X_{1}, X_{2}, \ldots , X_{n_{1}+n_{2}-n}$ on a continuous random variable $X$ with $-\infty < X < \infty$, is divided at random into three subsets of $(n_{1}-n)$, $(n_{2}-n)$, and $n$ observations $(n_{1}>n, n_{2}>n)$. Let $S_{1}, S_{2}$, and $S_{3}$ denote the corresponding sums. Find the characteristic function of

Show that the following distributions are infinitely divisible: (a) Pascal, (b) Cauchy, (c) Laplace, (d) Gamma.

The radius of a circle is approximately measured so that it has the uniform distribution in the interval $(a, b)$. Find the distribution:

Suppose that $X$ and $Y$ are independent random variables with the same exponential density

Show that the sum $X + Y$ and the ratio $X/Y$ are independent.

Suppose that $X$ and $Y$ are independent uniform variables in $(0, 1)$. Find the probability that the roots of $\lambda^{2} + 2X\lambda + Y = 0$ are real.

The angle $\varphi$ at which a projectile is fired with initial velocity $v$ follows the uniform distribution in the interval $(0, \pi /2)$. Find the distribution of the horizontal distance $d$ between the firing point and the point at which the projectile falls.

Suppose that $X$ and $Y$ are normal variables with mean $0$, variance $\sigma^{2}$, and correlation coefficient $\rho$.

Let $A = (X, Y)$ be the point of impact of a shot on a vertical target with center at origin $O$ and suppose the distribution of $A$ is that of Exercise 220 with $\rho = 0$ (called circular normal). Show that the distance $R = (OA)$ between the point of impact and the centre of the target and the angle of $OA$ with the horizontal axis (i.e., the polar coordinates of $A$) are independent variables. Find the distribution of $R^{2}$.

If $X$ has the uniform distribution on the interval $(0, 1)$, then:

If $X_{1}, X_{2}$ are independent and uniform on the interval $(0, 1)$, find the densities of:

Two friends $A$ and $B$ agree to meet between 12 (noon) and 1 p.m. at a restaurant. Supposing that they arrive at random between 12 and 1 p.m. independently of each other and the lunch lasts 30 minutes, what is the probability that they meet in the restaurant? Let $T$ be the instant of their meeting. What is the conditional distribution of $T$,

Let $X$ and $Y$ be independent with densities

Show that $XY$ is $\mathscr {N}\left(0, \sigma^{2}\right)$.

Given $n$ independent random numbers $X_{1}, X_{2}, \ldots , X_{n}$ from $(0, 1)$, called pseudorandom numbers, show that:

Let $X$ be a continuous random variable $X$ with distribution function $F(\cdot )$; consider the random variable

This transformation is called the probability integral transformation.

Show that the distribution of $Y$ is uniform in the interval $(0, 1)$, that is,

Given a random sample $X_{1}, X_{2}, \ldots , X_{n}$ from the uniform distribution, show that the solutions $y_{1}, y_{2}, \ldots , y_{n}$ of $y_{1} = F^{-1}(x_{1}), y_{2} = F^{-1}(x_{2}), \ldots , y_{n} = F^{-1}(x_{n})$ form a random sample from the $F$ distribution.

(Continuation of Problem 227.) Suppose the height $X$ of men has the normal distribution $\mathscr {N}\left(167 \text{ cm}, 9 \text{ cm}^{2}\right)$ and a random sample of 10 men is taken with heights $X_{1}, X_{2}, \ldots , X_{10}$. What is probability that the (random) interval $\delta = [\min (X_{1}, \ldots , X_{10}), \max (X_{1}, \ldots , X_{10})]$ covers at least 95 of the population of heights? That is, the $\mathbb {P}\left(\mathbb {P}\left(X \in \delta \right) \geq 0.95\right)$ is to be calculated; note that the $\mathbb {P}\left(X \in \delta \right)$, depending on the random interval $\delta$, is a random variable.

$X$ is called a lognormal variable, if the $\log X = Y$ has a normal distribution $\mathscr {N}\left(\mu , \sigma^{2}\right)$.

Show that if $F(x, y)$ is the distribution function of $X$ and $Y$ and

then

If $F(x, y)$ is continuous find the densities of $Z$ and $W$.

(Continuation of Problem 230.) If $X$ and $Y$ are independent $\mathscr {N}\left(0, 1\right)$, show that

Let $X = (X_{1}, X_{2}, \ldots , X_{k})$ have the Dirichlet distribution. Find the distribution,

(Continuation of Problem 232.) If $X_{i}$ $(i = 1, 2, \ldots , k+1)$ are independent Gamma variables with parameters $n_{1}, \ldots , n_{k+1}$ and $\lambda$, respectively, show that the

have the $k$-dimensional Dirichlet distribution (5.11).

Let $X$ and $Y$ be independent Poisson variables with parameters $\lambda$ and $\mu$. Show that:

Let $X$ and $Y$ be independent binomial with parameters $N, p$ and $M, p$, respectively. Show that:

Show that the mean

and the variance

of the random sample $X_{1}, X_{2}, \ldots , X_{n}$ from $\mathscr {N}\left(\mu , \sigma^{2}\right)$ are independent. Hence deduce the distribution of $(n-1)s^{2}$.

Hint:

Test for the Behrens–Fisher problem. To test whether two heteroscedastic normal populations have the same means one uses the statistic

where $\overline{x}, \overline{y}$ are the sample means of the independent samples from $\mathscr {N}\left(\mu_{1}, \sigma_{1}^{2}\right)$ and from $\mathscr {N}\left(\mu_{2}, \sigma_{2}^{2}\right)$, respectively, $m \leq n$ and $u_{j} = x_{j} - (\sqrt{m/n}) y_{j}$ $(j = 1, 2, \ldots , m)$. Show that $t$ has Student's distribution with $m-1$ degrees of freedom when $\mu_{1} = \mu_{2} = \mu$.