Mathematical Statistics and Data Analysis

John A. Rice

Chapter 2 Random Variables - all with Video Answers

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Chapter Questions

01:59

Problem 1

Suppose that $X$ is a discrete random variable with $P(X=0)=.25, P(X=1)=$ $.125, P(X=2)=.125,$ and $P(X=3)=.5 .$ Graph the frequency function and the cumulative distribution function of $X.$

Tony Wilson

Numerade Educator

02:04

An experiment consists of throwing a fair coin four times. Find the frequency function and the cumulative distribution function of the following random variables: (a) the number of heads before the first tail, (b) the number of heads following the first tail, (c) the number of heads minus the number of tails, and (d) the number of tails times the number of heads.

Nick Johnson

Numerade Educator

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Problem 3

The following table shows the cumulative distribution function of a discrete random variable. Find the frequency function.
$$\begin{array}{cc}
\hline k & F(k) \\
\hline 0 & 0 \\
1 & .1 \\
2 & .3 \\
3 & .7 \\
4 & .8 \\
5 & 1.0 \\
\hline
\end{array}$$

Donna Densmore

Numerade Educator

02:38

Problem 4

If $X$ is an integer-valued random variable, show that the frequency function is related to the cdf by $p(k)=F(k)-F(k-1).$

Sonam Khatri

Numerade Educator

04:06

Problem 5

Show that $P(u<X \leq v)=F(v)-F(u)$ for any $u$ and $v$ in the cases that (a) $X$ is a discrete random variable and (b) $X$ is a continuous random variable.

Amany Waheeb

Numerade Educator

05:25

Problem 6

Let $A$ and $B$ be events, and let $I_{A}$ and $I_{B}$ be the associated indicator random variables. Show that
$$I_{A \cap B}=I_{A} I_{B}=\min \left(I_{A}, I_{B}\right)$$
and
$$I_{A \cup B}=\max \left(I_{A}, I_{B}\right).$$

Mengchun Cai

Numerade Educator

01:34

Problem 7

Find the cdf of a Bernoulli random variable.

Wendi Zhao

Numerade Educator

05:09

Problem 8

Show that the binomial probabilities sum to 1.

Bryan Lynn

Numerade Educator

02:33

Problem 9

For what values of $p$ is a two-out-of-three majority decoder better than transmission of the message once?

Nick Johnson

Numerade Educator

03:01

Problem 10

Appending three extra bits to a 4 -bit word in a particular way (a Hamming code) allows detection and correction of up to one error in any of the bits. If each bit has probability .05 of being changed during communication, and the bits are changed independently of each other, what is the probability that the word is correctly received (that is, 0 or 1 bit is in error)? How does this probability compare to the probability that the word will be transmitted correctly with no check bits, in which case all four bits would have to be transmitted correctly for the word to be correct?

Dalia Rodriguez

Numerade Educator

07:05

Problem 11

Consider the binomial distribution with $n$ trials and probability $p$ of success on each trial. For what value of $k$ is $P(X=k)$ maximized? This value is called the mode of the distribution. (Hint: Consider the ratio of successive terms.)

Jacob Fry

Numerade Educator

01:47

Problem 12

Which is more likely: 9 heads in 10 tosses of a fair coin or 18 heads in 20 tosses?

Jacob Fry

Numerade Educator

01:28

Problem 13

A multiple-choice test consists of 20 items, each with four choices. A student is able to eliminate one of the choices on each question as incorrect and chooses randomly from the remaining three choices. A passing grade is 12 items or more correct.
a. What is the probability that the student passes?
b. Answer the question in part (a) again, assuming that the student can eliminate two of the choices on each question.

Manisha Sarker

Numerade Educator

03:16

Problem 14

Two boys play basketball in the following way. They take turns shooting and stop when a basket is made. Player A goes first and has probability $p_{1}$ of making a basket on any throw. Player $B$, who shoots second, has probability $p_{2}$ of making a basket. The outcomes of the successive trials are assumed to be independent.
a. Find the frequency function for the total number of attempts.
b. What is the probability that player A wins?

Narayan Hari

Numerade Educator

02:07

Problem 15

Two teams, $A$ and $B$, play a series of games. If team A has probability .4 of winning each game, is it to its advantage to play the best three out of five games or the best four out of seven? Assume the outcomes of successive games are independent.

Anas Venkitta

Numerade Educator

04:22

Problem 16

Show that if $n$ approaches $\infty$ and $r / n$ approaches $p$ and $m$ is fixed, the hypergeometric frequency function tends to the binomial frequency function. Give a heuristic argument for why this is true.

Aman Gupta

Numerade Educator

00:51

Problem 17

Suppose that in a sequence of independent Bernoulli trials, each with probability of success $p,$ the number of failures up to the first success is counted. What is the frequency function for this random variable?

Amany Waheeb

Numerade Educator

01:10

Problem 18

Continuing with Problem 17, find the frequency function for the number of failures up to the $r$ th success.

Chai Santi

Numerade Educator

01:13

Problem 19

Find an expression for the cumulative distribution function of a geometric random variable.

Matt Just

Numerade Educator

03:52

Problem 20

If $X$ is a geometric random variable with $p=.5,$ for what value of $k$ is $P(X \leq k) \approx .99 ?$

Sarah Gift

Numerade Educator

01:31

Problem 21

If $X$ is a geometric random variable, show that
$$P(X>n+k-1 | X>n-1)=P(X>k)$$
In light of the construction of a geometric distribution from a sequence of independent Bernoulli trials, how can this be interpreted so that it is "obvious"?

Hunza Gilgit

Numerade Educator

00:48

Problem 22

Three identical fair coins are thrown simultaneously until all three show the same face. What is the probability that they are thrown more than three times?

Sheryl Ezze

Numerade Educator

01:10

Problem 23

In a sequence of independent trials with probability $p$ of success, what is the probability that there are $r$ successes before the $k$th failure?

Wendi Zhao

Numerade Educator

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Problem 24

(Banach Match Problem) A pipe smoker carries one box of matches in his left pocket and one box in his right. Initially, each box contains $n$ matches. If he needs a match, the smoker is equally likely to choose either pocket. What is the frequency function for the number of matches in the other box when he first discovers that one box is empty?

Rashmi Sinha

Numerade Educator

01:03

Problem 25

The probability of being dealt a royal straight flush (ace, king, queen, jack, and ten of the same suit) in poker is about $1.3 \times 10^{-8} .$ Suppose that an avid poker player sees 100 hands a week, 52 weeks a year, for 20 years.
a. What is the probability that she is never dealt a royal straight flush dealt?
b. What is the probability that she is dealt exactly two royal straight flushes?

James Chok

Numerade Educator

07:17

Problem 26

The university administration assures a mathematician that he has only 1 chance in 10,000 of being trapped in a much-maligned elevator in the mathematics building. If he goes to work 5 days a week, 52 weeks a year, for 10 years, and always rides the elevator up to his office when he first arrives, what is the probability that he will never be trapped? That he will be trapped once? Twice? Assume that the outcomes on all the days are mutually independent (a dubious assumption in practice).

Ashley Boni

Numerade Educator

02:03

Problem 27

Suppose that a rare disease has an incidence of 1 in $1000 .$ Assuming that members of the population are affected independently, find the probability of $k$ cases in a population of 100,000 for $k=0, 1, 2.$

Sheryl Ezze

Numerade Educator

06:49

Problem 28

Let $p_{0}, p_{1}, \ldots, p_{n}$ denote the probability mass function of the binomial distribution with parameters $n$ and $p .$ Let $q=1-p .$ Show that the binomial probabilities can be computed recursively by $p_{0}=q^{n}$ and
$$p_{k}=\frac{(n-k+1) p}{k q} p_{k-1}, \quad k=1,2, \ldots, n$$
Use this relation to find $P(X \leq 4)$ for $n=9000$ and $p=.0005.$

Sheryl Ezze

Numerade Educator

02:52

Problem 29

Show that the Poisson probabilities $p_{0}, p_{1}, \ldots$ can be computed recursively by $p_{0}=\exp (-\lambda)$ and
$$p_{k}=\frac{\lambda}{k} p_{k-1}, \quad k=1,2, \ldots$$
Use this scheme to find $P(X \leq 4)$ for $\lambda=4.5$ and compare to the results of Problem 28.

Hunza Gilgit

Numerade Educator

02:28

Problem 30

Suppose that in a city, the number of suicides can be approximated by a Poisson process with $\lambda=.33$ per month.
a. Find the probability of $k$ suicides in a year for $k=0,1,2, \ldots .$ What is the most probable number of suicides?
b. What is the probability of two suicides in one week?

Sheryl Ezze

Numerade Educator

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Problem 31

Phone calls are received at a certain residence as a Poisson process with parameter $\lambda=2$ per hour.
a. If Diane takes a 10 -min. shower, what is the probability that the phone rings during that time?
b. How long can her shower be if she wishes the probability of receiving no phone calls to be at most .5?

Rashmi Sinha

Numerade Educator

01:26

Problem 32

For what value of $k$ is the Poisson frequency function with parameter $\lambda$ maximized? (Hint: Consider the ratio of consecutive terms.)

Hoan Nguyen

Numerade Educator

01:55

Problem 33

Let $F(x)=1-\exp \left(-\alpha x^{\beta}\right)$ for $x \geq 0, \alpha>0, \beta>0,$ and $F(x)=0$ for $x<0.$
Show that $F$ is a cdf, and find the corresponding density.

Michelle Z.

Numerade Educator

15:32

Problem 34

Let $f(x)=(1+\alpha x) / 2$ for $-1 \leq x \leq 1$ and $f(x)=0$ otherwise, where $-1 \leq \alpha \leq 1 .$ Show that $f$ is a density, and find the corresponding cdf. Find the quartiles and the median of the distribution in terms of $\alpha .$

Willis James

Numerade Educator

03:07

Problem 35

Sketch the pdf and cdf of a random variable that is uniform on $[-1,1] .$

Wendi Zhao

Numerade Educator

03:21

Problem 36

If $U$ is a uniform random variable on $[0,1],$ what is the distribution of the random variable $X=[n U],$ where $[t]$ denotes the greatest integer less than or equal to $t ?$

Amany Waheeb

Numerade Educator

01:16

Problem 37

A line segment of length 1 is cut once at random. What is the probability that the longer piece is more than twice the length of the shorter piece?

Manik Pulyani

Numerade Educator

02:23

Problem 38

If $f$ and $g$ are densities, show that $\alpha f+(1-\alpha) g$ is a density, where $0 \leq \alpha \leq 1.$

Jacob Fry

Numerade Educator

02:34

Problem 39

The Cauchy cumulative distribution function is
$$F(x)=\frac{1}{2}+\frac{1}{\pi} \tan ^{-1}(x), \quad-\infty<x<\infty$$
a. Show that this is a cdf.
b. Find the density function.
c. Find $x$ such that $P(X>x)=.1$

Neel Faucher

Numerade Educator

05:30

Problem 40

Suppose that $X$ has the density function $f(x)=c x^{2}$ for $0 \leq x \leq 1$ and $f(x)=0$ otherwise.
a. Find $c .$
b. Find the cdf.
c. What is $P(.1 \leq X<.5) ?$

Willis James

Numerade Educator

01:09

Problem 41

Find the upper and lower quartiles of the exponential distribution.

Hoan Nguyen

Numerade Educator

10:06

Problem 42

Find the probability density for the distance from an event to its nearest neighbor for a Poisson process in the plane.

Mengchun Cai

Numerade Educator

08:28

Problem 43

Find the probability density for the distance from an event to its nearest neighbor for a Poisson process in three-dimensional space.

Susan Hallstrom

Numerade Educator

01:25

Problem 44

Let $T$ be an exponential random variable with parameter $\lambda .$ Let $X$ be a discrete random variable defined as $X=k$ if $k \leq T<k+1, k=0,1, \ldots .$ Find the frequency function of $X.$

Amany Waheeb

Numerade Educator

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Problem 45

Suppose that the lifetime of an electronic component follows an exponential distribution with $\lambda=.1$
a. Find the probability that the lifetime is less than $10 .$
b. Find the probability that the lifetime is between 5 and $15 .$
c. Find $t$ such that the probability that the lifetime is greater than $t$ is .01

Rashmi Sinha

Numerade Educator

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Problem 46

$T$ is an exponential random variable, and $P(T<1)=.05 .$ What is $\lambda ?$

Rashmi Sinha

Numerade Educator

01:28

Problem 47

If $\alpha>1,$ show that the gamma density has a maximum at $(\alpha-1) / \lambda.$

Amany Waheeb

Numerade Educator

01:55

Problem 48

Show that the gamma density integrates to 1.

Amany Waheeb

Numerade Educator

16:56

Problem 49

The gamma function is a generalized factorial function.
a. Show that $\Gamma(1)=1$
b. Show that $\Gamma(x+1)=x \Gamma(x) .$ (Hint: Use integration by parts.)
c. Conclude that $\Gamma(n)=(n-1) !$, for $n=1,2,3, \ldots$
d. Use the fact that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ to show that, if $n$ is an odd integer,
$$\Gamma\left(\frac{n}{2}\right)=\frac{\sqrt{\pi}(n-1) !}{2^{n-1}\left(\frac{n-1}{2}\right) !}$$

Helen Delorenzo

Numerade Educator

02:25

Problem 50

Show by a change of variables that
$$\begin{aligned}
\Gamma(x) &=2 \int_{0}^{\infty} t^{2 x-1} e^{-t^{2}} d t \\
&=\int_{-\infty}^{\infty} e^{x t} e^{-e^{t}} d t.
\end{aligned}$$

Sajin Shajee

Numerade Educator

08:00

Problem 51

Show that the normal density integrates to $1 .$ (Hint: First make a change of variables to reduce the integral to that for the standard normal. The problem is then to show that $\int_{-\infty}^{\infty} \exp \left(-x^{2} / 2\right) d x=\sqrt{2 \pi} .$ Square both sides and reexpress the problem as that of showing
$$\left(\int_{-\infty}^{\infty} \exp \left(-x^{2} / 2\right) d x\right)\left(\int_{-\infty}^{\infty} \exp \left(-y^{2} / 2\right) d y\right)=2 \pi$$
Finally, write the product of integrals as a double integral and change to polar coordinates.)

Sarah Gift

Numerade Educator

02:12

Problem 52

Suppose that in a certain population, individuals' heights are approximately normally distributed with parameters $\mu=70$ and $\sigma=3$ in.
a. What proportion of the population is over 6 ft. tall?
b. What is the distribution of heights if they are expressed in centimeters? In meters?

Hossam Mohamed

Numerade Educator

02:35

Problem 53

Let $X$ be a normal random variable with $\mu=5$ and $\sigma=10 .$ Find (a) $P(X>10),$ (b) $P(-20<X<15),$ and (c) the value of $x$ such that $P(X>x)=.05$

Sheryl Ezze

Numerade Educator

00:18

Problem 54

If $X \sim N\left(\mu, \sigma^{2}\right),$ show that $P(|X-\mu| \leq .675 \sigma)=.5$

Sriparna Bhattacharjee

Numerade Educator

03:17

Problem 55

$X \sim N\left(\mu, \sigma^{2}\right),$ find the value of $c$ in terms of $\sigma$ such that $P(\mu-c \leq X \leq$ $\mu+c)=.95$

Narayan Hari

Numerade Educator

03:54

Problem 56

If $X \sim N\left(0, \sigma^{2}\right),$ find the density of $Y=|X|.$

Mengchun Cai

Numerade Educator

01:02

Problem 57

$X \sim N\left(\mu, \sigma^{2}\right)$ and $Y=a X+b,$ where $a<0,$ show that $Y \sim N\left(a \mu+b, a^{2} \sigma^{2}\right).$

Raj Bala

Numerade Educator

00:56

Problem 58

If $U$ is uniform on $[0,1],$ find the density function of $\sqrt{U}.$

Hoan Nguyen

Numerade Educator

06:23

Problem 59

If $U$ is uniform on $[-1,1],$ find the density function of $U^{2}.$

Mengchun Cai

Numerade Educator

03:54

Problem 60

Find the density function of $Y=e^{z},$ where $Z \sim N\left(\mu, \sigma^{2}\right) .$ This is called the lognormal density, since log $Y$ is normally distributed.

Mengchun Cai

Numerade Educator

01:38

Problem 61

Find the density of $c X$ when $X$ follows a gamma distribution. Show that only $\lambda$ is affected by such a transformation, which justifies calling $\lambda$ a scale parameter.

Kratika Bhadauria

Numerade Educator

02:36

Problem 62

Show that if $X$ has a density function $f_{X}$ and $Y=a X+b,$ then
$$f_{Y}(y)=\frac{1}{|a|} f_{X}\left(\frac{y-b}{a}\right)$$

Amany Waheeb

Numerade Educator

01:21

Problem 63

Suppose that $\Theta$ follows a uniform distribution on the interval $[-\pi / 2, \pi / 2] .$ Find the cdf and density of $tan \Theta.$

Manik Pulyani

Numerade Educator

00:23

Problem 64

A particle of mass $m$ has a random velocity, $V$, which is normally distributed with parameters $\mu=0$ and $\sigma .$ Find the density function of the kinetic energy, $E=\frac{1}{2} m V^{2}.$

Amy Jiang

Numerade Educator

01:46

Problem 65

How could random variables with the following density function be generated from a uniform random number generator?
$$f(x)=\frac{1+\alpha x}{2}, \quad-1 \leq x \leq 1, \quad-1 \leq \alpha \leq 1$$

Kratika Bhadauria

Numerade Educator

01:46

Problem 66

Let $f(x)=\alpha x^{-\alpha-1}$ for $x \geq 1$ and $f(x)=0$ otherwise, where $\alpha$ is a positive parameter. Show how to generate random variables from this density from a uniform random number generator.

Kratika Bhadauria

Numerade Educator

01:34

Problem 67

The Weibull cumulative distribution function is
$$F(x)=1-e^{-(x / \alpha)^{\beta}}, \quad x \geq 0, \quad \alpha>0, \quad \beta>0$$
a. Find the density function.
b. Show that if $W$ follows a Weibull distribution, then $X=(W / \alpha)^{\beta}$ follows an exponential distribution.
c. How could Weibull random variables be generated from a uniform random number generator?

Ameer Said

Numerade Educator

01:01

Problem 68

If the radius of a circle is an exponential random variable, find the density function of the area.

Raj Bala

Numerade Educator

01:01

Problem 69

If the radius of a sphere is an exponential random variable, find the density function of the volume.

Raj Bala

Numerade Educator

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Problem 70

Let $U$ be a uniform random variable. Find the density function of $V=U^{-\alpha}$ $\alpha>0 .$ Compare the rates of decrease of the tails of the densities as a function of $\alpha .$ Does the comparison make sense intuitively?

Victor Salazar

Numerade Educator

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Problem 71

This problem shows one way to generate discrete random variables from a uniform random number generator. Suppose that $F$ is the cdf of an integer-valued random variable; let $U$ be uniform on $[0,1] .$ Define a random variable $Y=k$ if $F(k-1)<U \leq F(k) .$ Show that $Y$ has cdf $F .$ Apply this result to show how to generate geometric random variables from uniform random variables.

Rashmi Sinha

Numerade Educator

03:35

Problem 72

One of the most commonly used (but not one of the best) methods of generating pseudorandom numbers is the linear congruential method, which works as follows. Let $x_{0}$ be an initial number (the "seed"). The sequence is generated recursively as
$$x_{n}=\left(a x_{n-1}+c\right) \bmod m$$
a. Choose values of $a, c,$ and $m,$ and try this out. Do the sequences "look" random?
b. Making good choices of $a, c,$ and $m$ involves both art and theory. The following are some values that have been proposed: $(1) a=69069, c=0, m=2^{31}; (2) a=65539, c=0, m=2^{31}$. The latter is an infamous generator called RANDU. Try out these schemes, and examine the results.

Bryan Lynn

Numerade Educator