A First Course in Probability

Sheldon M. Ross

Chapter 6

Jointly Distributed Random Variables - all with Video Answers

Educators

Section 1

Problems

Problem 1

Two fair dice are rolled. Find the joint probability mass function of $X$ and $Y$ when
(a) $X$ is the largest value obtained on any die and $Y$ is the sum of the values;
(b) $X$ is the value on the first die and $Y$ is the larger of the two values;
(c) $X$ is the smallest and $Y$ is the largest value obtained on the dice.

Aishwarya Krishnakumar

Aishwarya Krishnakumar

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Problem 2

Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let $X_{i}$ equal 1 if the $i$ th ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of
(a) $X_{1}, X_{2}$
(b) $X_{1}, X_{2}, X_{3}$.

Amany Waheeb

Numerade Educator

Problem 3

In Problem 2, suppose that the white balls are numbered, and let $Y_{i}$ equal 1 if the $i$ th white ball is selected and 0 otherwise. Find the joint probability mass function of
(a) $Y_{1}, Y_{2}$
(b) $Y_{1}, Y_{2}, Y_{3}$.

Amany Waheeb

Numerade Educator

Problem 4

Repeat Problem 2 when the ball selected is replaced in the urn before the next selection.

Amany Waheeb

Numerade Educator

Problem 5

Repeat Problem 3a when the ball selected is replaced in the urn before the next selection.

Amany Waheeb

Numerade Educator

Problem 6

A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by $N_{1}$ the number of tests made until the first defective is spotted and by $N_{2}$ the number of additional tests until the second defective is spotted; find the joint probability mass function of $N_{1}$ and $N_{2}$.

Amany Waheeb

Numerade Educator

Problem 7

Consider a sequence of independent Bernoulli trials, each of which is a success with probability $p$. Let $X_{1}$ be the number of failures preceding the first success, and let $X_{2}$ be the number of failures between the first two successes. Find the joint mass function of $X_{1}$ and $X_{2}$

Amany Waheeb

Numerade Educator

Problem 8

The joint probability density function of $X$ and $Y$ is given by
$$
f(x, y)=c\left(y^{2} \div x^{2}\right) e^{-y} \quad-y \leq x \leq y, 0<y<\infty
$$
(a) Find $c$.
(b) Find the marginal densities of $X$ and $Y$.

Amany Waheeb

Numerade Educator

Problem 9

The joint probability density function of $X$ and $Y$ is given by
$$
f(x, y)=\frac{6}{7}\left(x^{2}+\frac{x y}{2}\right) \quad 0<x<1,0<y<2
$$
(a) Verify that this is indeed a joint density function.
(b) Compute the density function of $X$.
(c) Find $P\{X>Y\}$.
Chapter 6 Jointly Distributed Random Variables
(d) Find $P\left\{Y>\frac{1}{2} \mid X<\frac{1}{2}\right\}$.
(e) Find $E[X]$.

Amany Waheeb

Numerade Educator

Problem 10

The joint probability density function of $X$ and $Y$ is given b
$$
f(x, y)=e^{-(x+y)} \quad 0 \leq x<\infty, 0 \leq y<\infty
$$
Find (a) $P\{X<Y\}$ and (b) $P\{X<a\}$.

Amany Waheeb

Numerade Educator

Problem 11

A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 15 percent will purchase a color television set, and 40 percent will just be browsing. If 5 customers enter his store on a given day, what is the probability that he will sell exactly 2 ordinary sets and 1 color set on that day?

Amany Waheeb

Numerade Educator

Problem 12

The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter $\lambda=10 .$ Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?

Amany Waheeb

Numerade Educator

Problem 13

A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between $12: 15$ and $12: 45$ and if the woman independently arrives at a time uniformly distributed between 12:00 and 1 P.M., find the probability that the first to arrive waits no longer than 5 minutes. What is the probability that the man arrives first?

Amany Waheeb

Numerade Educator

Problem 14

An ambulance travels back and forth, at a constant speed, along a road of length $L$. At a certain moment of time an accident occurs at a point uniformly distributed on the road. [That is, its distance from one of the fixed ends of the road is uniformly distributed over $(0, L)$.] Assuming that the ambulance's location at the moment of the accident is also uniformly distributed, compute, assuming independence, the distribution of its distance from the accident.

Amany Waheeb

Numerade Educator

Problem 15

An ambulance travels back and forth, at a constant speed, along a road of length $L$. At a certain moment of time an accident occurs at a point uniformly distributed on the road. [That is, its distance from one of the fixed ends of the road is uniformly distributed over $(0, L)$.] Assuming that the ambulance's location at the moment of the accident is also uniformly distributed, compute, assuming independence, the distribution of its distance from the accident.

Amany Waheeb

Numerade Educator

Problem 16

Suppose that $n$ points are independently chosen at random on the perimet of a circle, and we want the probability that they all lie in some semicircl
Problems $\quad \mathbf{2 9 5}$
(That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line.)

Let $P_{1}, \ldots, P_{n}$ denote the $n$ points. Let $A$ denote the event that all the points are contained in some semicircle, and let $A_{i}$ be the event that all the points lie in the semicircle beginning at the point $P_{t}$ and going clockwise for $180^{\circ}$, $i=1, \ldots, n .$
(a) Express $A$ in terms of the $A_{i}$.
(b) Are the $A_{i}$ mutually exclusive?
(c) Find $P(A)$.

Amany Waheeb

Numerade Educator

Problem 17

Three points $X_{1}, X_{2}, X_{3}$ are selected at random on a line $L$. What is the probability that $X_{2}$ lies between $X_{1}$ and $X_{3}$ ?

Amany Waheeb

Numerade Educator

Problem 18

Two points are selected randomly on a line of length $L$ so as to be on opposite sides of the midpoint of the line. [In other words, the two points $X$ and $Y$ are independent random variables such that $X$ is uniformly distributed over (0, $L / 2$ ) and $Y$ is uniformly distributed over $(L / 2, L) .]$ Find the probability that the distance between the two points is greater than $L / 3$.

Amany Waheeb

Numerade Educator

Problem 19

Two points are selected randomly on a line of length $L$ so as to be on opposite sides of the midpoint of the line. [In other words, the two points $X$ and $Y$ are independent random variables such that $X$ is uniformly distributed over (0, $L / 2$ ) and $Y$ is uniformly distributed over $(L / 2, L) .]$ Find the probability that the distance between the two points is greater than $L / 3$.

Amany Waheeb

Numerade Educator

Problem 20

The joint density of $X$ and $Y$ is given by
$$
f(x, y)= \begin{cases}x e^{-(x+y)} & x>0, y>0 \\ 0 & \text { otherwise }\end{cases}
$$
Are $X$ and $Y$ independent? What if $f(x, y)$ were given by
$$
f(x, y)= \begin{cases}2 & 0<x<y, 0<y<1 \\ 0 & \text { otherwise }\end{cases}
$$

Amany Waheeb

Numerade Educator

Problem 21

Let
$$
f(x, y)=24 x y \quad 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq x+y \leq 1
$$
and let it equal 0 otherwise.(a) Show that $f(x, y)$ is a joint probability density function.
(b) Find $E[X]$.
(c) Find $E[Y]$.

Amany Waheeb

Numerade Educator

Problem 22

The joint density function of $X$ and $Y$ is
$$
f(x, y)= \begin{cases}x+y & 0<x<1,0<y<1 \\ 0 & \text { otherwise }\end{cases}
$$
(a) Are $X$ and $Y$ independent?
(b) Find the density function of $X$.
(c) Find $P\{X+Y<1\}$.

Amany Waheeb

Numerade Educator

Problem 23

The random variables $X$ and $Y$ have joint density function.
$$
f(x, y)=12 x y(1-x) \quad 0<x<1,0<y<1
$$
and equal to 0 otherwise.
(a) Are $X$ and $Y$ independent?
(b) Find $E[X]$.
(c) Find $E[Y]$.
(d) Find $\operatorname{Var}(X)$.
(e) Find $\operatorname{Var}(Y)$.

Amany Waheeb

Numerade Educator

Problem 24

The random variables $X$ and $Y$ have joint density function.
$$
f(x, y)=12 x y(1-x) \quad 0<x<1,0<y<1
$$
and equal to 0 otherwise.
(a) Are $X$ and $Y$ independent?
(b) Find $E[X]$.
(c) Find $E[Y]$.
(d) Find $\operatorname{Var}(X)$.
(e) Find $\operatorname{Var}(Y)$.

Amany Waheeb

Numerade Educator

Problem 25

Suppose that $10^{6}$ people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over $\left(0,10^{6}\right)$. Let $N$ denote the number that arrive in the first hour. Find an approximation for $P\{N=i\}$

Amany Waheeb

Numerade Educator

Problem 26

Suppose that $10^{6}$ people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over $\left(0,10^{6}\right)$. Let $N$ denote the number that arrive in the first hour. Find an approximation for $P\{N=i\}$

Amany Waheeb

Numerade Educator

Problem 27

If $X$ is uniformly distributed over $(0,1)$ and $Y$ is exponentially distributed with parameter $\dot{\lambda}=1$, find the distribution of (a) $Z=X+Y$ and (b) $Z=$ $X / Y$. Assume independence.

Amany Waheeb

Numerade Educator

Problem 28

If $X_{1}$ and $X_{2}$ are independent exponential random variables with respective parameters $\bar{\lambda}_{1}$ and $\lambda_{2}$, find the disfribution of $Z=X_{1} / X_{2} .$ Also compute $P\left\{X_{1}<X_{2}\right\}$

Amany Waheeb

Numerade Educator

Problem 29

If $X_{1}$ and $X_{2}$ are independent exponential random variables with respective parameters $\bar{\lambda}_{1}$ and $\lambda_{2}$, find the disfribution of $Z=X_{1} / X_{2} .$ Also compute $P\left\{X_{1}<X_{2}\right\}$

Amany Waheeb

Numerade Educator

Problem 30

The expected number of typographical errors on a page of a certain magazine is $.2$. What is the probability that an article of 10 pages contains (a) 0 , and (b) 2 or more typographical errors? Explain your reasoning!

Amany Waheeb

Numerade Educator

Problem 31

The monthly worldwide average number of airplane crashes of commercial airlines is $2.2$. What is the probability that there will be
(a) more than 2 such accidents in the next month;
(b) more than 4 such accidents in the next 2 months;
(c) more than 5 such accidents in the next 3 months? Explain your reasoning!

Amany Waheeb

Numerade Educator

Problem 32

The monthly worldwide average number of airplane crashes of commercial airlines is $2.2$. What is the probability that there will be
(a) more than 2 such accidents in the next month;
(b) more than 4 such accidents in the next 2 months;
(c) more than 5 such accidents in the next 3 months? Explain your reasoning!

Amany Waheeb

Numerade Educator

Problem 33

Jill's bowling scores are approximately normally distributed with mean 170 and standard deviation 20 , while Jack's scores are approximately normally distributed with mean 160 and standard deviation 15. If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that
(a). Jack's score is higher;
(b) the total of their scores is above 350 .

Amany Waheeb

Numerade Educator

Problem 34

According to the U.S. National Center for Health Statistics, $25.2$ percent of males and $23.6$ percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that
(a) at least 110 of these 400 people never eat breakfast;
(b) the number of the women who never eat breakfast is at least as large as the number of the men who never eat breakfast.

Amany Waheeb

Numerade Educator

Problem 35

In Problem 2 , calculate the conditional probability mass function of $X_{1}$ given that
(a) $X_{2}=1$;
(b) $X_{2}=0$.

Amany Waheeb

Numerade Educator

Problem 36

In Problem 4 , calculate the conditional probability mass function of $X_{1}$ given that
(a) $X_{2}=1$
(b) $X_{2}=0$.

Amany Waheeb

Numerade Educator

Problem 37

In Problem 3 , calculate the conditional probability mass function of $Y_{1}$ given that
(a) $Y_{2}=1$;
(b) $Y_{2}=0$.

Amany Waheeb

Numerade Educator

Problem 38

In Problem 5 , calculate the conditional probability mass function of $Y_{1}$ given that
(a) $Y_{2}=1$;
(b) $Y_{2}=0$.

Amany Waheeb

Numerade Educator

Problem 39

Choose a number $X$ at random from the set of numbers $\{1,2,3,4,5\}$. Now choose a number at random from the subset no larger than $X$, that is, from $\{1, \ldots, X]$. Call this second number $Y$.
(a) Find the joint mass function of $X$ and $Y$.
(b) Find the conditional mass function of $X$ given that $Y=i$. Do it for $i=1,2,3,4,5 .$
(c) Are $X$ and $Y$ independent? Why?

Ekaveera Kumar

Numerade Educator

Problem 40

Two dice are rolled. Let $X$ and $Y$ denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of $Y$ given $X=i$, for $i=1,2, \ldots, 6 .$ Are $X$ and $Y$ independent? Why?

Ekaveera Kumar

Numerade Educator

Problem 41

Two dice are rolled. Let $X$ and $Y$ denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of $Y$ given $X=i$, for $i=1,2, \ldots, 6 .$ Are $X$ and $Y$ independent? Why?

Ekaveera Kumar

Numerade Educator

Problem 42

The joint probability mass function of $X$ and $Y$ is given by
$$
\begin{array}{ll}
p(1,1)=\frac{1}{8} & p(1,2)=\frac{1}{4} \\
p(2,1)=\frac{1}{8} & p(2,2)=\frac{1}{2}
\end{array}
$$
(a) Compute the conditional mass function of $X$ given $Y=i, i=1,2$.
(b) Are $X$ and $Y$ independent?
(c) Compute $P\{X Y \leq 3\}, P\{X+Y>2\}, P\{X / Y>1\}$.

Ekaveera Kumar

Numerade Educator

Problem 43

The joint density function of $X$ and $Y$ is given by
$$
f(x, y)=x e^{-x(y+1)} \quad x>0, y>0
$$
(a) Find the conditional density of $X$, given $Y=y$, and that of $Y$, given $X=x$.
(b) Find the-density function of $Z=X Y$.

Ekaveera Kumar

Numerade Educator

Problem 44

The joint density of $X$ and $Y$ is
$$
f(x, y)=c\left(x^{2}-y^{2}\right) e^{-x} \quad 0 \leq x<\infty,-x \leq y \leq x
$$
Finel the conditional distribution of $Y$, given $X=x$.

Ekaveera Kumar

Numerade Educator

Problem 45

If $X_{1}, X_{2}, X_{3}$ are independent random variables that are uniformly distributed over $(a, b)$, compute the probability that the largest of the three is greater than the sum of the other two.

Amany Waheeb

Numerade Educator

Problem 46

A complex machine is able to operate effectively as long as at least 3 of its 5 motors are functioning. If each motor independently functions for a random amount of time with density function $f(x)=x e^{-x}, x>0$, compute the density function of the length of time that the machine functions.

Amany Waheeb

Numerade Educator

Problem 47

If 3 trucks break down at points randomly distributed on a road of length $L$, find the probability that no 2 of the trucks are within a distance $d$ of each other when $d \leq L / 2$.

Mahnoor Khan

Numerade Educator

Problem 48

Consider a sample of size 5 from a uniform distribution over $(0,1)$. Comput the probability that the median is in the interval $\left(\frac{1}{4}, \frac{3}{4}\right)$.

Amany Waheeb

Numerade Educator

Problem 49

If $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}$ are independent and identically distributed exponential random variables with the parameter $\lambda$, compute
(a) $P\left\{\min \left(X_{1}, \ldots, X_{5}\right) \leq a\right\}$;
(b) $P\left\{\max \left(X_{1}, \ldots, X_{5}\right) \leq a\right\}$.

Ekaveera Kumar

Numerade Educator

Problem 50

Derive the distribution of the range of a sample of size 2 from a distribut having density function $f(x)=2 x, 0<x<1$.

Ekaveera Kumar

Numerade Educator

Problem 51

Derive the distribution of the range of a sample of size 2 from a distribut having density function $f(x)=2 x, 0<x<1$.

Ekaveera Kumar

Numerade Educator

Problem 52

If $X$ and $Y$ are independent random variables both uniformly distributed over $(0,1)$, find the joint density function of $R=\sqrt{X^{2}+Y^{2}}, \Theta=\tan ^{-1} Y / X$.

Mahnoor Khan

Numerade Educator

Problem 53

If $U$ is uniform on $(0,2 \pi)$ and $Z$, independent of $U$, is exponential with rate 1 , show directly (without using. the results of Example $7 \mathrm{~b}$ ) that $X$ and $Y$ defined by
$$
\begin{aligned}
X &=\sqrt{2 Z} \cos U \\
Y &=\sqrt{2 Z} \sin U
\end{aligned}
$$
are independent unit normal random variables.

Victor Salazar

Numerade Educator

Problem 54

If $X$ and $Y$ have joint density function
$$
f(x, y)=\frac{1}{x^{2} y^{2}} \quad x \geq 1, y \geq 1
$$
(a) Compute the joint density function of $U=X Y, V=X / Y$.
(b) What are the marginal densities?

Ekaveera Kumar

Numerade Educator

Problem 55

If $X$ and $Y$ are independent and identically distributed uniform random variables on $(0,1)$, compute the joint density of
(a) $U=X+Y, V=X / Y$
Chapter 6 Jointly Distributed Random Variables
(b) $U=X, V=X / Y$
(c) $U=X+Y, V=X /(X+Y)$

Ekaveera Kumar

Numerade Educator

Problem 56

Repeat Problem 55 when $X$ and $Y$ are independent exponential random variables, each with parameter $\lambda=1$.

Amany Waheeb

Numerade Educator

Problem 57

Repeat Problem 55 when $X$ and $Y$ are independent exponential random variables, each with parameter $\lambda=1$.

Amany Waheeb

Numerade Educator

Problem 58

If $X_{i}$ and $X_{2}$ are independent exponential random variables each having parameter $\lambda$, find the joint density function of $Y_{1}=X_{1}+X_{2}$ and $Y_{2}=$ $e^{x_{i}}$

Ekaveera Kumar

Numerade Educator

Problem 59

If $X, Y$, and $Z$ are independent random variables having identical density functions $f(x)=e^{-x}, 0<x<\infty$, derive the joint distribution of $U=$ $X+Y, V=X+Z, W=Y+Z$

Amany Waheeb

Numerade Educator

Problem 60

Consider an urri containing $n$ balls, numbered $1, \ldots, n$, and suppose that $k$ of them are randomly withdrawn. Let $X_{i}$ equal 1 if ball numbered $i$ is removed and let it be 0 otherwise. Show that $X_{1}, \ldots, X_{n}$ are exchangeable.

Amany Waheeb

Numerade Educator