Problems

Let $X$ be continuous with density function $f(x) = C\left(x-x^{2}\right)$, where $\alpha < x < \beta$ and $C > 0$.

Let $X$ be a random variable which takes non-negative values only. Show that

where $A_{i} = \left\{ i-1 \leq X < i\right\}$. Deduce that

Show that $X$ and $Y$ are independent continuous variables if and only if their joint density function $f$ factorizes as the product $f(x, y) = g(x) h(y)$ of functions of the single variables $x$ and $y$ alone.

Explain why the density function $f(x, y) = C e^{-x-y}$ (for $x, y \geq 0, x+y > 1$) does not provide a counterexample to this assertion.

Let $X$ and $Y$ have joint density function $f(x, y) = 2 e^{-x-y}, 0 < x < y < \infty$. Are they independent? Find their marginal density functions and their covariance.

A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base can fit inside the circle if:

It is required to estimate $J = \int_{0}^{1} g(x) d x$ where $0 \leq g(x) \leq 1$ for all $x$, as in Example (2.6.3). Let $X$ and $Y$ be independent random variables with common density function $f(x) = 1$ if $0 < x < 1, f(x) = 0$ otherwise. Let $U = I_{\left\{ Y \leq g(X)\right\} }$, the indicator function of the event that $Y \leq g(X)$, and let $V = g(X), W = \frac{1}{2}\left\{ g(X)+g(1-X)\right\}$. Show that $\mathbb {E}\left[U\right] = \mathbb {E}\left[V\right] = \mathbb {E}\left[W\right] = J$, and that $\operatorname {Var}\left[W\right] \leq \operatorname {Var}\left[V\right] \leq \operatorname {Var}\left[U\right]$, so that, of the three, $W$ is the most 'efficient' estimator of $J$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent exponential variables, parameter $\lambda$. Show by induction that $S = X_{1}+X_{2}+\cdots +X_{n}$ has the $\Gamma (\lambda , n)$ distribution.

Let $X$ and $Y$ be independent variables, $\Gamma (\lambda , m)$ and $\Gamma (\lambda , n)$ respectively.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent $N(0,1)$ variables.

Let $X$ and $Y$ have the bivariate normal distribution with means $\mu_{1}, \mu_{2}$, variances $\sigma_{1}^{2}, \sigma_{2}^{2}$, and correlation $\rho$. Show that

Let $X$ and $Y$ have joint density function $f$. Find the density function of $Y / X$.

Let $X$ and $Y$ be independent variables with common density function $f$. Show that $\tan^{-1}(Y / X)$ has the uniform distribution on $(-\frac{1}{2} \pi , \frac{1}{2} \pi )$ if and only if

Verify that this is valid if either $f$ is the $N(0,1)$ density function or $f(x) = a\left(1+x^{4}\right)^{-1}$ for some constant $a$.

Let $X$ and $Y$ be independent $N(0,1)$ variables, and think of $(X, Y)$ as a random point in the plane. Change to polar coordinates $(R, \Theta )$ given by $R^{2} = X^{2}+Y^{2}, \tan \Theta = Y / X$; show that $R^{2}$ is $\chi^{2}(2), \tan \Theta$ has the Cauchy distribution, and $R$ and $\Theta$ are independent. Find the density of $R$.

Find $\mathbb {E}\left[X^{2} / R^{2}\right]$ and

If $X$ and $Y$ are independent random variables, show that $U = \min \left\{ X, Y\right\}$ and $V = \max \left\{ X, Y\right\}$ have distribution functions

Let $X$ and $Y$ be independent exponential variables, parameter 1. Show that

Let $X$ and $Y$ be independent variables having the exponential distribution with parameters $\lambda$ and $\mu$ respectively. Let $U = \min \left\{ X, Y\right\} , V = \max \left\{ X, Y\right\}$, and $W = V-U$.

Let $X$ and $Y$ be independent non-negative random variables with continuous density functions on $(0, \infty )$.

Show that it cannot be the case that $U = X+Y$ where $U$ is uniformly distributed on $[0,1]$ and $X$ and $Y$ are independent and identically distributed. You should not assume that $X$ and $Y$ are continuous variables.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent identically distributed variables with a common density function $f$. Such a collection is called a random sample. For each $\omega \in \Omega$, arrange the sample values $X_{1}(\omega ), \ldots , X_{n}(\omega )$ in non-decreasing order $X_{(1)}(\omega ) \leq X_{(2)}(\omega ) \leq \cdots \leq X_{(n)}(\omega )$, where (1), (2), ,., (n) is a (random) permutation of $1,2, \ldots , n$. The new variables $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$ are called the order statistics. Show, by a symmetry argument, that the joint distribution function of the order statistics satisfies

where $L$ is given by

and $\mathbf{x} = \left(x_{1}, x_{2}, \ldots , x_{n}\right)$. Deduce that the joint density function of $X_{(1)}, \ldots , X_{(n)}$ is $g(\mathbf{y}) = n!L(\mathbf{y}) f\left(y_{1}\right) \cdots f\left(y_{n}\right)$.

Find the marginal density function of the $k$ th order statistic $X_{(k)}$ of a sample with size $n$ :

Find the joint density function of the order statistics of $n$ independent uniform variables on $[0, T]$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent and uniformly distributed on $[0,1]$, with order statistics $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$.

Let $X_{1}, X_{2}, X_{3}$ be independent variables with the uniform distribution on $[0,1]$. What is the probability that rods of lengths $X_{1}, X_{2}$, and $X_{3}$ may be used to make a triangle? Generalize your answer to $n$ rods used to form a polygon.

Let $X_{1}$ and $X_{2}$ be independent variables with the uniform distribution on $[0, 1]$. A stick of unit length is broken at points distance $X_{1}$ and $X_{2}$ from one of the ends. What is the probability that the three pieces may be used to make a triangle? Generalize your answer to a stick broken in $n$ places.

Let $X, Y$ be a pair of jointly continuous variables.

Let $Z$ be a random variable. Choose $X$ and $Y$ appropriately in the Cauchy-Schwarz (or Hölder) inequality to show that $g(p) = \log \mathbb {E}\left[\left|Z^{p}\right|\right]$ is a convex function of $p$ on the interval of values of $p$ such that $\mathbb {E}\left[\left|Z^{p}\right|\right] < \infty$. Deduce Lyapunov's inequality:

You have shown in particular that, if $Z$ has finite $r$ th moment, then $Z$ has finite $s$ th moment for all positive $s \leq r$.

Show more generally that, if $r_{1}, r_{2}, \ldots , r_{n}$ are real numbers such that $\mathbb {E}\left[\left|X^{r_{k}}\right|\right] < \infty$ and $s = n^{-1} \sum_{k = 1}^{n} r_{k}$, then

Show that, using the obvious notation, $\mathbb {E}\left[\mathbb {E}\left[X \mid Y, Z\right] \mid Y\right] = \mathbb {E}\left[X \mid Y\right]$.

Motor cars of unit length park randomly in a street in such a way that the centre of each car, in turn, is positioned uniformly at random in the space available to it. Let $m(x)$ be the expected number of cars which are able to park in a street of length $x$. Show that

It is possible to deduce that $m(x)$ is about as big as $\frac{3}{4} x$ when $x$ is large.

Buffon's needle revisited: Buffon's noodle.

In the excitement of calculating $\pi$, Mr Buffon (no relation) inadvertently swallows the needle and is X-rayed. If the needle exhibits no preference for direction in the gut, what is the distribution of the length of its image on the X-ray plate? If he swallowed Buffon's cross (see Exercise (4.5.3)) also, what would be the joint distribution of the lengths of the images of the two arms of the cross?

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent exponential variables with parameter $\lambda$, and let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ be their order statistics. Show that

are also independent and have the same joint distribution as the $X_{i}$.

Let $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$ be the order statistics of a family of independent variables with common continuous distribution function $F$. Show that

are independent and uniformly distributed on $[0,1]$. This is equivalent to Problem (4.14.33). Why?

You are permitted to inspect the $n$ prizes at a fête in a given order, at each stage either rejecting or accepting the prize under consideration. There is no recall, in the sense that no rejected prize may be accepted later. It may be assumed that, given complete information, the prizes may be ranked in a strict order of preference, and that the order of presentation is independent of this ranking. Find the strategy which maximizes the probability of accepting the best prize, and describe its behaviour when $n$ is large.

Let $R^{2} = X^{2}+Y^{2}+Z^{2}$ where $X, Y, Z$ are independent normal random variables with means $\lambda , \mu , \nu$, and common variance $\sigma^{2}$, where $(\lambda , \mu , \nu ) \neq (0,0,0)$. Show that the conditional density of the point $(X, Y, Z)$ given $R = r$, when expressed in spherical polar coordinates relative to an axis in the direction $\mathbf{e} = (\lambda , \mu , \nu )$, is of the form

where $a = r\left|\mathbf{e}\right|$.

Let $\phi$ be the $N(0,1)$ density function, and define the functions $H_{n}, n \geq 0$, by $H_{0} = 1$, and $(-1)^{n} H_{n} \phi = \phi^{(n)}$, the $n$th derivative of $\phi$. Show that:

Let $X$ and $Y$ have a standard bivariate normal distribution with zero means, unit variances, and correlation coefficient $\rho$, and suppose $U = u(X)$ and $V = v(Y)$ have finite variances. Show that $\left|\rho (U, V)\right| \leq \left|\rho \right|$. [Hint: Use Problem (4.14.37) to expand the functions $u$ and $v$. You may assume that $u$ and $v$ lie in the linear span of the $H_{n}$.]

Let $X_{(1)}, X_{(2)}, \ldots , X_{(n)}$ be the order statistics of $n$ independent random variables, uniform on $[0,1]$. Show that:

The six coordinates $\left(X_{i}, Y_{i}\right), 1 \leq i \leq 3$, of three points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ in the plane are independent $N(0,1)$. Show that the the probability that C lies inside the circle with diameter AB is $\frac{1}{4}$.

The coordinates $\left(X_{i}, Y_{i}, Z_{i}\right), 1 \leq i \leq 3$, of three points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are independent $N(0,1)$. Show that the probability that C lies inside the sphere with diameter AB is $\frac{1}{3}-\frac{\sqrt{3}}{4 \pi }$.

Let $X$ have variance $\sigma^{2}$ and write $m_{k} = \mathbb {E}\left[X^{k}\right]$. Define the skewness of $X$ by $\operatorname {skw}(X) = \mathbb {E}\left[\left(X-m_{1}\right)^{3}\right] / \sigma^{3}$. Show that:

Let $X$ have variance $\sigma^{2}$ and $\mathbb {E}\left[X^{k}\right] = m_{k}$. Define the kurtosis of $X$ by $\operatorname {kur}(X) = \mathbb {E}\left[\left(X-m_{1}\right)^{4}\right] / \sigma^{4}$. Show that:

Fisher-Gumbel-Tippett distribution. Let $X_{r}, 1 \leq r \leq n$, be independent and exponentially distributed with parameter 1. Show that $X_{(n)} = \max \left\{ X_{r}: 1 \leq r \leq n\right\}$ satisfies

Hence show that $\int_{0}^{\infty }\left\{ 1-\exp \left(-e^{-x}\right)-\exp \left(-e^{x}\right)\right\} d x = \gamma$ where $\gamma$ is Euler's constant.

Let $S$ and $X$ have density functions satisfying $b(x) \leq f_{S}(x) \leq a(x)$ and $f_{S}(x) \leq f_{X}(x)$. Let $U$ be uniformly distributed on $[0,1]$ and independent of $X$. Given the value $X$, we implement the following algorithm:

Show that, conditional on ultimate acceptance, $X$ is distributed as $S$. Explain when you might use this method of sampling.

Let $X, Y$, and $\left\{ U_{r}: r \geq 1\right\}$ be independent random variables, where:

and the $U_{r}$ are uniform on $[0,1]$. Let $M = \max \left\{ U_{1}, U_{2}, \ldots , U_{Y}\right\}$, and show that $Z = X-M$ is exponentially distributed.

Let $U$ and $V$ be independent and uniform on $[0,1]$. Set $X = -\alpha^{-1} \log U$ and $Y = -\log V$ where $\alpha > 0$.