EXERCISES Section 3.2 (11-28)

11

Let $X$ be the number of students who show up for a professor’s office hour on a particular day. Suppose that the pmf of $X$ is $p\left(0\right)=.20,p\left(1\right)=.25,p\left(2\right)=.30,p\left(3\right)=.15$ , and $p\left(4\right)=.10$ .

a. Draw the corresponding probability histogram.

b. What is the probability that at least two students show up? More than two students show up?

c. What is the probability that between one and three students, inclusive, show up?

d. What is the probability that the professor shows up?

12

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable $Y$ as the number of ticketed passengers who actually show up for the flight. The probability mass function of $Y$ appears in the accompanying table.

y	45	46	47	48	49	50	51	52	53	54	55
p(y)	.05	.10	.12	.14	.25	.17	.06	.05	.03	.02	.01

a. What is the probability that the flight will accommodate all ticketed passengers who show up?

b. What is the probability that not all ticketed passengers who show up can be accommodated?

c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

13

A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table.

x	0	1	2	3	4	5	6
p(x)	.10	.15	.20	.25	.20	.06	.04

Calculate the probability of each of the following events.

a. {at most three lines are in use }

b. {fewer than three lines are in use }

c. {at least three lines are in use }

d. {between two and five lines, inclusive, are in use }

e. {between two and four lines, inclusive, are not in use }

f. {at least four lines are not in use }

14

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let $Y=$ the number of forms required of the next applicant. The probability that $y$ forms are required is known to be proportional to $y$ -that is, $p\left(y\right)=ky$ for $y=1,\dots ,5$ .

a. What is the value of $k$ ? Hint: ${\sum }_{y=1}^{5}p\left(y\right)=1$

b. What is the probability that at most three forms are required?

c. What is the probability that between two and four forms (inclusive) are required?

d. Could $p\left(y\right)=y^2/50$ for $y=1,\dots ,5$ be the pmf of $Y$ ?

15

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair $\left(1,2\right)$ represents the selection of boards 1 and 2 for inspection.

a. List the ten different possible outcomes.

b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define $X$ to be the number of defective boards observed among those inspected. Find the probability distribution of $X$ .

c. Let $F\left(x\right)$ denote the cdf of $X$ . First determine $F\left(0\right)=$ $P\left(X\le 0\right),F\left(1\right)$ , and $F\left(2\right)$ ; then obtain $F\left(x\right)$ for all other $x$ .

16

Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area, $25\%$ of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let $X$ denote the number among the four who have earthquake insurance.

a. Find the probability distribution of $X$ . [Hint: Let $S$ denote a homeowner who has insurance and $F$ one who does not. Then one possible outcome is SFSS, with probability $\left(.25\right)\left(.75\right)\left(.25\right)\left(.25\right)$ and associated $X$ value 3. There are 15 other outcomes.]

b. Draw the corresponding probability histogram.

c. What is the most likely value for $X$ ?

d. What is the probability that at least two of the four selected have earthquake insurance?

17

A new battery’s voltage may be acceptable $\left(A\right)$ or unacceptable $\left(U\right)$ . A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that $90\%$ of all batteries have acceptable voltages. Let $Y$ denote the number of batteries that must be tested.

a. What is $p\left(2\right)$ , that is, $P\left(Y=2\right)$ ?

b. What is $p\left(3\right)$ ? [Hint: There are two different outcomes that result in $Y=3$ .]

c. To have $Y=5$ , what must be true of the fifth battery selected? List the four outcomes for which $Y=5$ and then determine $p\left(5\right)$ .

d. Use the pattern in your answers for parts (a)-(c) to obtain a general formula for $p\left(y\right)$ .

18

Two fair six-sided dice are tossed independently. Let $M=$ the maximum of the two tosses (so $M\left(1,5\right)=5$ , $M\left(3,3\right)=3$ , etc.).

a. What is the pmf of $M$ ? Hint: First determine $p\left(1\right)$ , then $p\left(2\right)$ , and so on.

b. Determine the cdf of $M$ and graph it.

19

A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesday’s mail. In actuality, each one may arrive on Wednesday, Thursday, Friday, or Saturday. Suppose the two arrive independently of one another, and for each one $P\left(\text{Wed.}\right)=.3$ , $P\left(\text{Thurs.}\right)=.4,P\left(\text{Fri.}\right)=.2$, and $P\left(\text{Sat.}\right)=.1$ . Let $Y=$ the number of days beyond Wednesday that it takes for both magazines to arrive (so possible $Y$ values are 0,1, 2, or 3). Compute the pmf of $Y$ . Hint: There are 16 possible outcomes; $Y\left(W,W\right)=0,Y\left(F,Th\right)=2$ , and so on.

20

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let $X=$ the number of people who arrive late for the seminar.

a. Determine the probability mass function of $X$ . Hint: label the three couples $\#1,\#2$ , and $\#3$ and the two individuals #4 and #5.

b. Obtain the cumulative distribution function of $X$ , and use it to calculate $P\left(2\le X\le 6\right)$ .

21

Suppose that you read through this year’s issues of the New York Times and record each number that appears in a news article-the income of a CEO, the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be $1,2,\dots ,8$ , or 9 . Your first thought might be that the leading digit $X$ of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution called Benford’s law:

$p\left(x\right)=P\left(1\text{ st digit is }x\right)={\log }_{10}\left(\frac{x+1}{x}\right)x=1,2,\dots ,9$

a. Without computing individual probabilities from this formula, show that it specifies a legitimate pmf.

b. Now compute the individual probabilities and compare to the corresponding discrete uniform distribution.

c. Obtain the cdf of $X$ .

d. Using the cdf, what is the probability that the leading digit is at most 3 ? At least 5 ?

Note: Benford’s law is the basis for some auditing procedures used to detect fraud in financial reporting-for example, by the Internal Revenue Service.

22

Refer to Exercise 13, and calculate and graph the cdf $F\left(x\right)$ . Then use it to calculate the probabilities of the events given in parts (a)-(d) of that problem.

23

A branch of a certain bank in New York City has six ATMs. Let $X$ represent the number of machines in use at a particular time of day. The cdf of $X$ is as follows:

$F\left(x\right)=\{\begin{array}{ll}0& x<0\\ .06& 0\le x<1\\ .19& 1\le x<2\\ .39& 2\le x<3\\ .67& 3\le x<4\\ .92& 4\le x<5\\ .97& 5\le x<6\\ 1& 6\le x\end{array}$

Calculate the following probabilities directly from the cdf:

a. $p\left(2\right)$ , that is, $P\left(X=2\right)$ b. $P\left(X>3\right)$ c. $P\left(2\le X\le 5\right)$ d. $P\left(2<X<5\right)$

24

An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let $X$ be the number of months between successive payments. The cdf of $X$ is as follows: $F\left(x\right)=\{\begin{array}{ll}0& x<1\\ .30& 1\le x<3\\ .40& 3\le x<4\\ .45& 4\le x<6\\ .60& 6\le x<12\\ 1& 12\le x\end{array}$

a. What is the pmf of $X$ ?

b. Using just the cdf, compute $P\left(3\le X\le 6\right)$ and $P\left(4\le X\right)$ .

25

In Example 3.12, let $Y=$ the number of girls born before the experiment terminates. With $p=P\left(B\right)$ and $1-p=P\left(G\right)$ , what is the pmf of $Y$ ? Hint: First list the possible values of $Y$ , starting with the smallest, and proceed until you see a general formula.

26

Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at $A,B,C$ , and $D$ . One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit. Once at a friend’s house, he will either return home or else proceed to one of the two adjacent houses (such as $0,A$ , or $C$ when at $B$ ), with each of the three possibilities having probability $1/3$ . In this way, Alvie continues to visit friends until he returns home.

a. Let $X=$ the number of times that Alvie visits a friend. Derive the pmf of $X$ .

b. Let $Y=$ the number of straight-line segments that Alvie traverses (including those leading to and from $0)$ . What is the pmf of $Y$ ?

c. Suppose that female friends live at $A$ and $C$ and male friends at $B$ and $D$ . If $Z=$ the number of visits to female friends, what is the pmf of $Z$ ?

27

After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books in a completely random fashion to each of the four students $\left(1,2,3\text{,and 4}\right)$ who claim to have left books. One possible outcome is that 1 receives 2’s book, 2 receives 4’s book, 3 receives his or her own book, and 4 receives 1’s book. This outcome can be abbreviated as $\left(2,4,3,1\right)$ .

a. List the other 23 possible outcomes.

b. Let $X$ denote the number of students who receive their own book. Determine the pmf of $X$ .

28

Show that the cdf $F\left(x\right)$ is a nondecreasing function; that is, $x_1<x_2$ implies that $F\left(x_1\right)\le F\left(x_2\right)$ . Under what condition will $F\left(x_1\right)=F\left(x_2\right)$ ?