General Exercises

Using the Cauchy--Schwarz inequality show that

is a convex function. Then show that $[\mathbb {E}\left[\left|X\right|^{t}\right]]^{1/t}$ is an increasing function of $t$.

Every real characteristic function $\varphi_{0}(t)$ satisfies the inequality

Every real characteristic function $\varphi_{0}(t)$ satisfies

If $X_{1}, X_{2}, X_{3}, X_{4}$ are jointly normal with $\operatorname {Cov}\left[ X_{i}, X_{j} \right] = \mathbb {E}\left[X_{i} X_{j}\right] = \sigma_{ij}$, show by the use of moment generating functions, that

Let $X_{1}, X_{2}, \ldots , X_{n}$ be a random sample from the uniform distribution in $(0, 1)$ and let $Y_{n} = n \min (X_{1}, X_{2}, \ldots , X_{n})$. Using characteristic functions, show that the asymptotic distribution ($n \to \infty$) of $Y_{n}$ is the exponential $e^{-y}$, $y > 0$.

Let $(X_{1}, Y_{1}), \ldots , (X_{n}, Y_{n})$ be a random sample from the bivariate normal with means 0, variances 1, and correlation $\rho$. Let the random variable

Find $\mathbb {E}\left[\overline{Z}\right]$.

$n$ tickets are numbered 1 through $n$. Of these, $r$ tickets drawn at random are the lucky ones. A lucky ticket receives an amount equal to the number it bears. What is the expected amount given on every draw? What is the variance of this amount?

In the generalized binomial with $\mathbb {P}\left(X_{i} = 1\right) = p_{i}$ ($i = 1, 2, \ldots , n$), show that as $n \to \infty$ the distribution of the number of successes $X = \sum_{i=1}^{n} X_{i}$ tends to the Poisson with parameter $\lambda$ where $\lambda = \lim_{n \to \infty } \sum_{i=1}^{n} p_{i}$ assuming $\max_{i} \lambda_{i} \to 0$.

Let $X$ be the number of failures preceding the $n$th success in a sequence of independent Bernoulli trials with probability of success $p$. Show that if as $n \to \infty$, $q = 1-p \to 0$, so that $nq \to \lambda$, then

If the continuous random variable $X$ is such that for some integer $k > 0$, $kX$ has the Gamma distribution and the conditional distribution of $Y$ given $X$ is Poisson, then $Y$ has a negative binomial distribution.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be a random sample from the Laplace distribution

By using characteristic functions show that the random variable

is distributed as $(\sigma /2n)\chi^{2}_{2n}$. Hence deduce that $\mathbb {E}\left[Y\right] = \sigma$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be a random sample from a distribution with $\mathbb {E}\left[X_{i}\right] = 0$ and $\operatorname {Var}\left[X_{i}\right] = 1$. Show that

where $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$.

Let $X_{(1)}, \ldots , X_{(n)}$ be the ordered sample from a uniform distribution in $(0, 1)$ and $Y_{r} = X_{(1)} + X_{(2)} + \cdots + X_{(r)}$. Show that the conditional distribution of $W = Y_{r}/Y_{(r+1)}$ is the distribution of the sum of $r$ observations of a random sample from the same uniform distribution.

$N$ points $X_{1}, X_{2}, \ldots , X_{n}$ are selected at random in $(0, 1)$. Show that the distribution of the sum $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$ has density

where the sum extends as long as $x-1, x-2, \ldots$ and $(n-1-x), (n-2-x), \ldots$ are positive.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent discrete variables with the same uniform distribution on the set of integers $0, 1, \ldots , m-1$, i.e.,

Find the probability generating function of the sum $S_{n} = X_{1} + \cdots + X_{n}$ and hence the probability function of $S_{n}$.

(Continuation of Problem 315.) A die is thrown three times. What is the probability that the sum of the outcomes is 10?

A chain smoker has $m+1$ match boxes; each time he lights a cigarette he chooses one of the boxes at random. Suppose that (to begin with) every box contains $N$ matches. Let $X_{1}, X_{2}, \ldots , X_{m}$ be the numbers of matches left in the remaining $m$ boxes, when:

(Continuation of Problem 317.) Suppose that box $C_{k}$ ($k = 1, \ldots , m$) is chosen with probability $p_{k}$. What is the probability that $C_{k}$ empties first?

Show that $X_{1}, X_{2}, X_{3}$ with a joint discrete distribution are independent if and only if

for all $(x_{1}, x_{2})$ with $\mathbb {P}\left(X_{1} = x_{1}, X_{2} = x_{2}\right) > 0$ and

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent and identically distributed positive random variables with $\mathbb {E}\left[X_{i}\right] = \mu$, $\mathbb {E}\left[X_{i}^{-1}\right] = r$. Let $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$. Show that $\mathbb {E}\left[S_{n}^{-1}\right]$ exists and

(Continuation of Problem 320.) Show that

A caterpillar is moving on the edges of a tetrahedron $ABCD$ on whose top there is glue. In a unit of time the caterpillar goes from any vertex (except $D$) to any other vertex with the same probability $1/3$. Suppose that the caterpillar at time $t = 0$ is on the vertex $A$. What is the probability that

A flea moves at random on a plane with leaps of constant length $a$ and in a random direction at each step. If it starts from the origin, let $(X_{n}, Y_{n})$ be the position of the flea after $n$ jumps. Where

The angles $\theta_{i}$ are independently uniform in $(0, 2\pi )$.

Let the random variables $X_{1}, \ldots , X_{n}$ have the joint density

Show that any $n-1$ of the $X_{1}, \ldots , X_{n}$ are independent normal with mean 0 and variance 1. Are the $X_{1}, \ldots , X_{n}$ jointly normal?

In Exercise 288 suppose the plane is divided into squares by equidistant lines parallel to the axes. Let $2a$ be the side of each square and $2\mu$ the length of the needle. Find the average number of lines crossed by the needle,

Let $X, Y$ be independent $\chi^{2}$ variables each with $n$ degrees of freedom. Show that

has the Student distribution with $n$ degrees of freedom. Hence deduce that:

If for a given value $\lambda$ of $\Lambda$ the random variable $X(\Lambda )$ is Poisson with parameter $\lambda$ and $\Lambda$ is a random variable with a gamma density

show that $X(\Lambda )$ has the negative binomial distribution

where $p = (1+a)^{-1}$, $q = a(a+1)^{-1}$.

Show that for any two random variable(s) $X$ and $Y$ with $\operatorname {Var}\left[X\right] < \infty$,

Using the distribution of $X_{(t)}$ where $X_{(1)}, \ldots , X_{(n)}$ is the ordered sample from the uniform in $(0, 1)$, show the following relation between the incomplete $B$ function $I_{x}(n_{1}, n_{2})$ and the binomial distribution function: