Probability: An Introduction

Geoffrey Grimmett, Dominic Welsh

Chapter 6 Multivariate distributions and independence - all with Video Answers

Educators

Chapter Questions

Problem 1

If $X$ and $Y$ are independent random variables with density functions $f_{X}$ and $f_{Y}$, respectively, show that $U=X Y$ and $V=X / Y$ have density functions
$$
f_{U}(u)=\int_{-\infty}^{\infty} f_{X}(x) f_{Y}(u / x) \frac{1}{|x|} d x, \quad f_{V}(v)=\int_{-\infty}^{\infty} f_{X}(v y) f_{Y}(y)|y| d y
$$

Amany Waheeb

Numerade Educator

03:53

Problem 2

Is the function $G$, defined by
$$
G(x, y)= \begin{cases}1 & \text { if } x+y \geq 0 \\ 0 & \text { otherwise }\end{cases}
$$
the joint distribution function of some pair of random variables? Justify your answer.

Amany Waheeb

Numerade Educator

View

Problem 3

Let $(X, Y, Z)$ be a point chosen uniformly at random in the unit cube $(0,1)^{3}$. Find the probability that the quadratic equation $X t^{2}+Y t+Z=0$ has two distinct real roots.

Rashmi Sinha

Numerade Educator

03:00

Problem 4

Show that if $X$ and $Y$ are independent random variables having the exponential distribution with parameters $\lambda$ and $\mu$, respectively, then $\min \{X, Y\}$ has the exponential distribution with parameter $\lambda+\mu$.

Nick Johnson

Numerade Educator

04:28

Problem 5

Lack-of-memory property. If $X$ has the exponential distribution, show that
$$
\mathbb{P}(X>u+v \mid X>u)=\mathbb{P}(X>v) \quad \text { for } u, v>0
$$
This is called the 'lack of memory' property, since it says that, if we are given that $X>u$, then the distribution of $X-u$ is the same as the original distribution of $X$. Show that if $Y$ is a positive, continuous random variable with the lack-of-memory property above, then $Y$ has the exponential distribution.

Amany Waheeb

Numerade Educator

01:01

Problem 6

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, each having distribution function $F$ and density function $f$. Find the distribution function of $U$ and the density functions of $U$ and $V$, where $U=\min \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}$ and $V=\max \left\{X_{1}, X_{2}, \ldots, X_{n}\right\}$. Show that the joint density function of $U$ and $V$ is
$$
f_{U, V}(u, v)=n(n-1) f(u) f(v)[F(v)-F(u)]^{n-2} \quad \text { if } u<v
$$

Dominador Tan

Numerade Educator

02:36

Problem 7

Let $X_{1}, X_{2}, \ldots$ be independent, identically distributed, continuous random variables. Define $N$ as the index such that
$$
X_{1} \geq X_{2} \geq \cdots \geq X_{N-1} \quad \text { and } \quad X_{N-1}<X_{N}
$$
Prove that $\mathbb{P}(N=k)=(k-1) / k !$ and that $\mathbb{E}(N)=e$.

Amany Waheeb

Numerade Educator

View

Problem 8

Show that there exists a constant $c$ such that the function
$$
f(x, y)=\frac{c}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \quad \text { for } x, y \in \mathbb{R}
$$
is a joint density function. Show that both marginal density functions of $f$ are the density function of the Cauchy distribution.

Victor Salazar

Numerade Educator

View

Problem 9

Let $X$ and $Y$ have joint density function
$$
f(x, y)= \begin{cases}\frac{1}{4}(x+3 y) e^{-(x+y)} & \text { if } x, y \geq 0 \\ 0 & \text { otherwise }\end{cases}
$$
Find the marginal density function of $Y$. Show that $\mathbb{P}(Y>X)=\frac{5}{8}$.

Victor Salazar

Numerade Educator

03:10

Problem 10

Let $S_{n}$ be the sum of $n$ independent, identically distributed random variables having the exponential distribution with parameter $\lambda$. Show that $S_{n}$ has the gamma distribution with parameters $n$ and $\lambda$.
For given $t>0$, show that $N_{t}=\max \left\{n: S_{n} \leq t\right\}$ has a Poisson distribution.

Jacob Fry

Numerade Educator

07:25

Problem 11

An aeroplane drops medical supplies to two duellists. With respect to Cartesian coordinates whose origin is at the target point, both the $x$ and $y$ coordinates of the landing point of the supplies have normal distributions which are independent. These two distributions have the same mean 0 and variance $\sigma^{2}$. Show that the expectation of the distance between the landing point and the target is $\sigma \sqrt{\pi / 2}$. What is the variance of this distance? (Oxford $1976 \mathrm{M}$ )

Willis James

Numerade Educator

01:18

Problem 12

$X$ and $Y$ are independent random variables normally distributed with mean zero and variance $\sigma^{2}$. Find the expectation of $\sqrt{X^{2}+Y^{2}}$. Find the probabilities of the following events, where $a, b, c$, and $\alpha$ are positive constants such that $b<c$ and $\alpha<\frac{1}{2} \pi$ :
(a) $\sqrt{X^{2}+Y^{2}}<a$
(b) $0<\tan ^{-1}(Y / X)<\alpha$ and $Y>0$.
(Consider various cases depending on the relative sizes of $a, b$, and $c$.) (Oxford 1981M)

Manik Pulyani

Numerade Educator

03:00

Problem 13

The independent random variables $X$ and $Y$ are both exponentially distributed with parameter $\lambda$, that is, each has density function
$$
f(t)= \begin{cases}\lambda e^{-\lambda t} & \text { if } t>0 \\ 0 & \text { otherwise }\end{cases}
$$
(a) Find the (cumulative) distribution and density functions of the random variables $1-e^{-\lambda X}$, $\min \{X, Y\}$, and $X-Y$.
(b) Find the probability that $\max \{X, Y\} \leq a X$, where $a$ is a real constant. (Oxford 1982M)

Nick Johnson

Numerade Educator

01:41

Problem 14

The independent random variables $X$ and $Y$ are normally distributed with mean 0 and variance 1
(a) Show that $W=2 X-Y$ is normally distributed, and find its mean and variance.
(b) Find the mean of $Z=X^{2} /\left(X^{2}+Y^{2}\right)$.
(c) Find the mean of $V / U$, where $U=\max \{|X|,|Y|\}$ and $V=\min \{|X|,|Y|\}$.
(Oxford 1985M)

Kari Hasz

Numerade Educator

04:33

Problem 15

Let $X$ and $Y$ be independent random variables, $X$ having the normal distribution with mean 0 and variance 1 , and $Y$ having the $\chi^{2}$ distribution with $n$ degrees of freedom. Show that
$$
T=\frac{X}{\sqrt{Y / n}}
$$
has density function
$$
f(t)=\frac{1}{\sqrt{\pi n}} \frac{\Gamma\left(\frac{1}{2}(n+1)\right)}{\Gamma\left(\frac{1}{2} n\right)}\left(1+\frac{t^{2}}{n}\right)^{-\frac{1}{2}(n+1)} \quad \text { for } t \in \mathbb{R}
$$
$T$ is said to have the $t$-distribution with $n$ degrees of freedom.

Philomena Marfo

Numerade Educator

04:33

Problem 16

Let $X$ and $Y$ be independent random variables with the $\chi^{2}$ distribution, $X$ having $m$ degrees of freedom and $Y$ having $n$ degrees of freedom. Show that
$$
U=\frac{X / m}{Y / n}
$$
has density function
$$
f(u)=\frac{m \Gamma\left(\frac{1}{2}(m+n)\right)}{n \Gamma\left(\frac{1}{2} m\right) \Gamma\left(\frac{1}{2} n\right)} \cdot \frac{(m u / n)^{\frac{1}{2} m-1}}{[1+(m u / n)]^{\frac{1}{2}(m+n)}} \quad \text { for } u>0
$$
$U$ is said to have the $F$-distribution with $m$ and $n$ degrees of freedom.

Philomena Marfo

Numerade Educator

03:05

Problem 17

In a sequence of dependent Bernoulli trials, the conditional probability of success at the $i$ th trial, given that all preceding trials have resulted in failure, is $p_{i}(i=1,2, \ldots)$. Give an expression in terms of the $p_{i}$ for the probability that the first success occurs at the $n$th trial.
Suppose that $p_{i}=1 /(i+1)$ and that the time intervals between successive trials are independent random variables, the interval between the $(n-1)$ th and the $n$th trials being exponentially distributed with density $n^{\alpha} \exp \left(-n^{\alpha} x\right)$, where $\alpha$ is a given constant. Show that the expected time to achieve the first success is finite if and only if $\alpha>0$. (Oxford $1975 \mathrm{~F}$ )

Amany Waheeb

Numerade Educator

03:44

Problem 18

Let $a, b>0$. Independent positive random variables $X$ and $Y$ have probability densities
$$
\frac{1}{\Gamma(a)} x^{a-1} e^{-x}, \quad \frac{1}{\Gamma(b)} y^{b-1} e^{-y}, \quad \text { for } x, y \geq 0
$$
respectively, and $U$ and $V$ are defined by
$$
U=X+Y, \quad V=\frac{X}{X+Y}
$$
Prove that $U$ and $V$ are independent, and find their distributions.
Deduce that
$$
\mathbb{E}\left(\frac{X}{X+Y}\right)=\frac{\mathbb{E}(X)}{\mathbb{E}(X)+\mathbb{E}(Y)}
$$

James Kiss

Numerade Educator

04:33

Problem 19

Let $X_{1}, X_{2}, X_{3}$ be independent $\chi^{2}$ random variables with $r_{1}, r_{2}, r_{3}$ degrees of freedom.
(a) Show that $Y_{1}=X_{1} / X_{2}$ and $Y_{2}=X_{1}+X_{2}$ are independent and that $Y_{2}$ is a $\chi^{2}$ random variable with $r_{1}+r_{2}$ degrees of freedom.
(b) Deduce that the following random variables are independent:
$$
\frac{X_{1} / r_{1}}{X_{2} / r_{2}} \quad \text { and } \quad \frac{X_{3} / r_{3}}{\left(X_{1}+X_{2}\right) /\left(r_{1}+r_{2}\right)}
$$

Philomena Marfo

Numerade Educator

05:31

Problem 20

Let $X$ and $Y$ be random variables with the vector $(X, Y)$ uniformly distributed on the region $R=\{(x, y): 0<y<x<1\}$. Write down the joint probability density function of $(X, Y)$ Find $\mathbb{P}(X+Y<1)$.

Find the probability density function $f_{X}(x)$ of $X$, and find also $\mathbb{E}(X)$. Find the conditional probability density function $f_{Y \mid X}(y \mid x)$ of $Y$ given that $X=x$, and find also $\mathbb{E}(Y \mid X=x)$.

Willis James

Numerade Educator

01:29

Problem 21

Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Let $U=$ $\min \{U, V\}$ and $V=\max \{U, V\}$. Show that $\mathbb{E}(U)=\frac{1}{3}$, and hence find the covariance of $U$ and $V$. (Cambridge 2007)

Nick Johnson

Numerade Educator

05:28

Problem 22

Three crew members of Dr Who's spacecraft Tardis are teleported to the surface of the spherical planet $Z o g .$ Their positions $X, Y, Z$ are independent and uniformly distributed on the surface. Find the probability density function of the angle $\widehat{X C Y}$, where $C$ is the centre of $\mathrm{Zog}$. Two people positioned on the surface at $A$ and $B$ are in direct radio communication if and only if $\widehat{A C B}<\frac{1}{2} \pi$
(a) Find the probability that $Z$ is in direct radio communication with either $X$ or $Y$, conditional on the event that $\phi:=\widehat{X C Y}$ satisfies $\phi<\frac{1}{2} \pi$.
(b) Find the probability that $Z$ is in direct radio communication with both $X$ and $Y$, conditional on the event that $\phi>\frac{1}{2} \pi$.
Deduce that the probability that all three crew members can keep in touch is $(\pi+2) /(4 \pi)$.

Sheryl Ezze

Numerade Educator

00:47

Problem 23

Zog continued. This time, $n$ members of Dr Who's crew are transported to Zog, their positions being independent and uniformly distributed on the surface. In addition, Dr Who is required to choose a place $W$ on the surface for his own transportation. Find the probability that, for every $W$, he is able to communicate with some member of his crew.

Harsh Gadhiya

Numerade Educator

03:58

Problem 24

Let $X$ and $Y$ be independent non-negative random variables with densities $f$ and $g$, respectively. Find the joint density function of $U=X$ and $V=X+a Y$, where $a$ is a positive constant.

Let $X$ and $Y$ be independent and exponentially distributed random variables, each with density
$$
f(x)=\lambda e^{-\lambda x} \quad \text { for } x \geq 0
$$
Find the density of $X+\frac{1}{2} Y$. Is it the same as the density of $\max \{X, Y\} ?$ (Cambridge 2007)

Amany Waheeb

Numerade Educator

01:29

Problem 25

Let $X$ and $Y$ have the bivariate normal density function
$$
f(x, y)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}} \exp \left\{-\frac{1}{2\left(1-\rho^{2}\right)}\left(x^{2}-2 \rho x y+y^{2}\right)\right\} \quad \text { for } x, y \in \mathbb{R}
$$
for fixed $\rho \in(-1,1)$. Let $Z=(Y-\rho X) / \sqrt{1-\rho^{2}}$. Show that $X$ and $Z$ are independent $\mathrm{N}(0,1)$ variables. Hence or otherwise determine $\mathbb{P}(X>0, Y>0)$. (Cambridge 2008)

Hoan Nguyen

Numerade Educator

01:59

Problem 26

Let $X$ and $Y$ be random variables with the joint probability density function
$$
f_{X, Y}(x, y)=\frac{1}{4} e^{-\frac{1}{2}(x+y)} \quad \text { for } x, y>0
$$
Show that the joint probability density function of $U=\frac{1}{2}(X-Y)$ and $V=Y$ is
$$
f_{U, V}(u, v)= \begin{cases}\frac{1}{2} e^{-u-v} & \text { if }(u, v) \in A \\ 0 & \text { otherwise }\end{cases}
$$
where $A$ is a region of the ( $u, v)$ plane to be determined. Deduce that $U$ has probability density function
$$
f_{U}(u)=\frac{1}{2} e^{-|u|}, \quad-\infty<u<\infty
$$

Nick Johnson

Numerade Educator

01:59

Problem 27

(a) Suppose that the continuous random variables $X$ and $Y$ are independent with probability density functions $f$ and $g$, both of which are symmetric about zero.
(i) Find the joint probability density function of $(U, V)$, where $U=X$ and $V=Y / X$.
(ii) Show that the marginal density function of $V$ is
$$
f_{V}(v)=2 \int_{0}^{\infty} x f(x) g(x v) d x
$$
(iii) Let $X$ and $Y$ be independent normal random variables, each with mean 0 , and with non-zero variances $a^{2}$ and $b^{2}$, respectively. Show that $V=Y / X$ has probability density function
$$
f_{V}(v)=\frac{c}{\pi\left(c^{2}+v^{2}\right)} \quad \text { for }-\infty<v<\infty
$$
where $c=b / a$. Hence find $\mathbb{P}(|Y|<|X|)$.
(b) Now let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $(0,1)$. By considering the random variables $U=Y$ and $V=X Y^{2}$, or otherwise, find the probability density function of $V$.

Nick Johnson

Numerade Educator

12:30

Problem 28

(a) Define the distribution function $F$ of a random variable, and also its density function $f$, assuming $F$ is differentiable. Show that
$$
f(x)=-\frac{d}{d x} \mathbb{P}(X>x)
$$
(b) Let $U, V$ be independent random variables, each with the uniform distribution on $[0,1]$. Show that
$$
\mathbb{P}\left(V^{2}>U>x\right)=\frac{1}{3}-x+\frac{2}{3} x^{3 / 2} \quad \text { for } x \in(0,1)
$$
(c) What is the probability that the random quadratic equation $x^{2}+2 V x+U=0$ has real roots?
(d) Given that the two roots $R_{1}, R_{2}$ of the above quadratic are real, what is the probability that both $\left|R_{1}\right| \leq 1$ and $\left|R_{2}\right| \leq 1 ?$

Amany Waheeb

Numerade Educator