Chapter 11

In Exercises 1 and 2, the outcomes and corresponding probability assignments for a discrete random variable \(X\) are listed. Draw the histogram for \(X\). Then find the expected value \(E(X)\), the variance \(\operatorname{Var}(X)\), and the standard deviation \(\sigma(X)\). $$ \begin{array}{l|c|c|c|c|c} \hline \text { Outcomes for } X & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & \frac{1}{9} & \frac{2}{9} & \frac{1}{3} & \frac{1}{9} & \frac{2}{9} \\ \hline \end{array} $$

In Exercises 1 and 2, the outcomes and corresponding probability assignments for a discrete random variable \(X\) are listed. Draw the histogram for \(X\). Then find the expected value \(E(X)\), the variance \(\operatorname{Var}(X)\), and the standard deviation \(\sigma(X)\). $$ \begin{array}{l|c|c|c|c|c} \hline \text { Outcomes for } X & 0 & 2 & 4 & 6 & 8 \\ \hline \text { Probability } & \frac{1}{8} & \frac{3}{8} & \frac{1}{4} & \frac{1}{8} & \frac{1}{8} \\ \hline \end{array} $$

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) counts the number of eggs laid by a randomly selected fruit fly.

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) measures the annual distance flown by a randomly selected airplane from a particular airline.

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) measures the annual salary of a randomly selected player on a particular major league baseball team.

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) counts the number of books in the library of a randomly selected professor at your school.

In Exercises 7 through \(10, f(x)\) is a probability density function for a particular continuous random variable \(X\). In each case, find the indicated probabilities. $$ \begin{aligned} &f(x)= \begin{cases}\frac{1}{5} & \text { if } 3 \leq x \leq 8 \\ 0 & \text { otherwise }\end{cases} \\ &P(2 \leq X \leq 7) \text { and } P(X \geq 5) \end{aligned} $$

In Exercises 7 through \(10, f(x)\) is a probability density function for a particular continuous random variable \(X\). In each case, find the indicated probabilities. $$ \begin{aligned} &f(x)= \begin{cases}\frac{1}{x^{2}} & \text { if } x \geq 1 \\ 0 & \text { if } x<1\end{cases} \\ &P(1 \leq X \leq 3) \text { and } P(X \geq 2) \end{aligned} $$

In Exercises 7 through \(10, f(x)\) is a probability density function for a particular continuous random variable \(X\). In each case, find the indicated probabilities. $$ \begin{aligned} &f(x)= \begin{cases}0.25 x e^{-x / 2} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{cases} \\ &P(X \geq 2) \text { and } P(0 \leq X \leq 3) \end{aligned} $$

Find a number \(c\) so that the following function \(f(x)\) is a probability density function: $$ f(x)= \begin{cases}c x e^{-x / 4} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{cases} $$

Find a number \(c\) so that the following function \(f(x)\) is a probability density function: $$ f(x)= \begin{cases}\frac{c}{x^{4}} & \text { if } x \geq 1 \\ 0 & \text { otherwise }\end{cases} $$

If the random variable \(X\) is normally distributed with mean \(\mu=7\) and standard deviation \(\sigma=2\), what is \(P(X \geq 9)\) ?

Find \(b\) if \(P(Z \geq b)=0.73\), where \(Z\) is a random variable with a standard normal distribution \((\mu=0, \sigma=1)\).

QUALITY CONTROL A toy manufacturer makes hollow rubber balls. The thickness of the outer shell of such a ball is normally distributed with mean \(0.03\) millimeter and standard deviation \(0.0015\) millimeter. What is the probability that the outer shell of a randomly selected ball will be less than \(0.025\) millimeter thick?

EFFECT OF AN EPIDEMIC A study of the effect of an epidemic of mononucleosis (mono) on the students at a particular small private college determines that during the 30 days of November, there were: 5 days when no students had mono 7 days when exactly one student had mono 4 days when exactly two students had mono 9 days when exactly six students had mono 3 days when exactly seven students had mono 2 days when exactly eight students had mono Let \(X\) be the random variable that measures the number of students with mononucleosis on a randomly selected day in November. a. Find the probability distribution for \(X\). Then construct a histogram for this distribution. b. How many students would you expect to have mononucleosis on a randomly selected day in November?

WHEEL OF FORTUNE A wheel of fortune is divided into 20 circular sectors of equal area. The wheel is spun and a payoff is made according to the color of the region on which the indicator lands. One region is gold and pays \(\$ 50\) when hit; five regions are blue and pay \(\$ 25\); and four other regions are red and pay \(\$ 10\). The remaining 10 regions are black and pay nothing. Would you be willing to pay \(\$ 10\) to play this game?

PEDIATRICS There are 200 children in a certain school, and the weight of the children is a random variable \(X\) that is normally distributed with mean \(\mu=80\) pounds and standard deviation \(\sigma=7\) pounds. a. How many children weigh more than 90 pounds? b. How many children weigh less than 70 pounds? c. How many children weigh exactly 80 pounds?

PRODUCT RELIABILITY Electronic components made by a certain process have a time to failure that is measured by a normal random variable. If the mean time to failure is 15,000 hours with a standard deviation of 800 hours, what is the probability that a randomly selected component will last no more than 10,000 hours?

LOTTERY The probability of winning \(\$ 100\) in a particular lottery is \(0.08\), the probability of winning \(\$ 20\) is \(0.12\), the probability of winning \(\$ 5\) is \(0.2\), and the probability of losing is \(0.6\). What is a fair price to pay for a lottery ticket?

TIME MANAGEMENT A bakery turns out a fresh batch of chocolate chip cookies every 45 minutes. Tina arrives (at random) at the bakery, hoping to buy a fresh cookie. Use an appropriate uniform density function to find the probability that Tina arrives within 5 minutes (before or after) the time that the cookies come out of the oven.

DEMOGRAPHICS A study recently commissioned by the mayor of a large city indicates that the number of years a current resident will continue to live in the city may be modeled as an exponential random variable with probability density function $$ f(t)= \begin{cases}0.4 e^{-0.4 t} & \text { for } t \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ a. Find the probability that a randomly selected resident will move within 10 years. b. Find the probability that a randomly selected resident will remain in the city for more than 20 years. c. How long should a randomly selected resident be expected to remain in town?

TRAFFIC CONTROL Suppose the time (in minutes) between the arrivals of successive cars at a toll booth is measured by the random variable \(X\) with probability density function $$ f(t)= \begin{cases}0.5 e^{-0.5 t} & \text { if } t \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ a. Find the probability that a randomly selected pair of successive cars will arrive at the toll booth at least 6 minutes apart. b. Find the average time between the arrivals of successive cars at the toll booth.

TRAFFIC MANAGEMENT The distance (in feet) between successive cars on a freeway is modeled by the random variable \(X\) with probability density function $$ f(x)= \begin{cases}0.25 x e^{-x / 2} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ a. Find the probability that a randomly selected pair of cars will be less than 10 feet apart. b. What is the average distance between successive cars on the freeway?

TRAFFIC MANAGEMENT Suppose the random variable \(X\) in Exercise 27 is normally distributed with mean \(\mu=12\) feet and standard deviation \(\sigma=4\) feet. Now what is the probability that a randomly selected pair of cars will be less than 10 feet apart?

INSURANCE POLICY An insurance company charges \(\$ 10,000\) for a policy insuring against a certain kind of accident and pays \(\$ 100,000\) if the accident occurs. Suppose it is estimated that the probability of the accident occurring is \(p=0.02\). Let \(X\) be the random variable that measures the insurance company's profit on each policy it sells. a. What is the probability distribution for \(X\) ? b. What is the company's expected profit per policy sold? c. What should the company charge per policy to double its expected profit per policy?

PERSONAL HEALTH Jules decides to go on a diet for 6 weeks, with a goal of losing between 10 and 15 pounds. Based on his body configuration and metabolism, his doctor determines that the amount of weight he will lose can be modeled by a continuous random variable \(X\) with probability density function \(f(x)\) of the form \(f(x)= \begin{cases}k(x-10)^{2} & \text { for } 10 \leq x \leq 15 \\ 0 & \text { otherwise }\end{cases}\) If the doctor's model is valid, how much weight should Jules expect to lose? [Hint: First determine the value of the constant \(k\).]

FISHERY MANAGEMENT Brooke, the manager of a fishery, determines that the age \(X\) (in weeks) at which a certain species of fish dies follows an exponential distribution with probability density function $$ f(t)= \begin{cases}\lambda e^{-\lambda x} & \text { for } t \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ Brooke observes that it is twice as likely for a randomly selected fish to die during the first 10 -week period as during the next 10 weeks (from week 10 to week 20 ). a. What is \(\lambda\) ? b. What is the probability that a randomly chosen fish will die within the first 5 weeks? c. How long should Brooke expect a randomly selected fish to live?

METALLURGY The proportion of impurities by weight in samples of copper ore taken from a particular mine is measured by a random variable \(X\) with probability density function \(f(x)= \begin{cases}21 x^{2}(1-\sqrt{x}) & \text { for } 0 \leq x \leq 1 \\ 0 & \text { otherwise }\end{cases}\) a. What is the probability that the proportion of impurities in a randomly selected sample will be less than \(5 \%\) ? b. What is the probability that the proportion of impurities will be greater than \(50 \%\) ? c. What proportion of impurities would you expect to find in a randomly selected sample?

BEVERAGES Suppose that the volume of soda in a bottle produced at a particular plant is normally distributed with a mean of 12 ounces and a standard deviation of \(0.05\) ounce. a. Find the probability that a bottle filled at this plant contains at least \(11.8\) ounces. b. Find the volume of soda so that \(95 \%\) of all bottles filled at this plant contain less than this amount.

QUALITY CONTROL An automobile manufacturer claims that its new cars get an average of 30 miles per gallon in city driving. Assume the manufacturer's claim is correct and that gas mileage is normally distributed, with standard deviation of 2 miles per gallon. a. Find the probability that a randomly selected car will get less than 25 miles per gallon. b. If you test two cars, what is the probability that both get less than 25 miles per gallon?

ACADEMIC TESTING The results of a calculus exam are normally distributed with a mean of \(72.3\) and a standard deviation of \(16.4\). Find the probability that a randomly chosen student's score is between 50 and 75 . If there are 82 students in the class, about how many have scores between 50 and \(75 ?\)

MEDICINE Suppose that the number of children who die each year from leukemia follows a Poisson distribution and that on average, \(7.3\) children per 100,000 die from leukemia. For a city with 100,000 children, find the probability of each of the following events: a. Exactly seven children in the city die from leukemia each year. b. Fewer than two children in the city die from leukemia each year. c. More than five children in the city die from leukemia each year.

ECOLOGY The pH level of a liquid measures its acidity and is an important issue in studying the effects of acid rain. Suppose that a test is conducted under controlled conditions that allow the change in \(\mathrm{pH}\) in a particular lake resulting from acid rain to be recorded. Let \(X\) be a random variable that measures the \(\mathrm{pH}\) of a sample of water taken from the lake, and assume that \(X\) has the probability density function. \(f(x)= \begin{cases}0.75(x-4)(6-x) & \text { for } 4 \leq x \leq 6 \\ 0 & \text { otherwise }\end{cases}\) a. Find the probability that the \(\mathrm{pH}\) of a randomly selected sample will be at least \(5 .\) b. Find the expected \(\mathrm{pH}\) of a randomly selected sample.

SPORTS MEDICINE Suppose that the number of injuries a team suffers during a typical football game follows a Poisson distribution with an average of \(2.5\) injuries. a. Find the probability that during a randomly chosen game, the team suffers exactly two injuries. b. Find the probability that during a randomly chosen game, the team suffers no injuries. c. Find the probability that during a randomly chosen game, the team suffers at least one injury.

PACKAGE DELIVERY Suppose that the number of overnight packages a business receives during a business day follows a Poisson distribution and that, on average, the company receives four overnight packages per day.| a. Find the probability that the company receives exactly four overnight packages on a randomly selected business day. b. Find the probability that the company does not receive any overnight packages on a randomly selected business day. c. Find the probability that the company receives fewer than four overnight packages on a randomly selected business day.

JOURNALISM Suppose that the number of typographical errors on a page of a local newspaper follows a Poisson distribution with an average of \(2.5\) errors per page. a. Find the probability that a randomly selected page is free from typographical errors. b. Find the probability that a randomly selected page has at least one typographical error. c. Find the probability that a randomly selected page has at least three typographical errors. d. Find the probability that a randomly selected page has fewer than three typographical errors.

HIGHWAY ACCIDENTS A report models the number of automobile accidents on a particular highway as a random variable with a Poisson distribution. Suppose it is found that on average, there is an accident every 10 hours. a. Find the probability that there are no accidents on this highway during a randomly selected 24-hour period. b. Find the probability that there is at least one accident on this highway during a randomly selected 12 -hour period. c. Find the probability that there are no accidents on this highway during a randomly selected hour.

PUBLIC HEALTH As part of a campaign to combat a new strain of influenza, public health authorities are planning to inoculate 1 million people. It is estimated that the probability of an individual having a bad reaction to the vaccine is \(0.0005\). Suppose the number of people inoculated who have bad reactions to the vaccine is modeled by a random variable with a Poisson distribution. a. What is \(\lambda\) for the distribution? b. What is the probability that of the 1 million people inoculated, exactly five will have a bad reaction? c. What is the probability that of the 1 million people inoculated more than 10 will have a bad reaction?

LABOR EFFICIENCY A company wishes to examine the efficiency of two members of its senior staff, Jack and Jill, who work independently of one another. Let \(X\) and \(Y\) be random variables that measure the proportion of the work week that Jack and Jill, respectively, actually spend performing their duties. Assume that the joint probability density function for \(X\) and \(Y\) is \(f(x, y)= \begin{cases}0.4(2 x+3 y) & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\ 0 & \text { otherwise }\end{cases}\) a. Verify that \(f(x, y)\) satisfies the requirements for a joint probability density function. b. Find the probability that Jack spends less than half his time working while Jill spends more than half her time working. c. Find the probability that Jack and Jill each spend at least \(80 \%\) of the work week performing their assigned tasks. d. Find the probability that Jack and Jill combine for less than a full work week. [Hint: This is the event that \(X+Y<1\).]

TIME MANAGEMENT Suppose the random variable \(X\) measures the time (in minutes) that a person stands in line at a certain bank and \(Y\) measures the duration (in minutes) of a routine transaction at the teller's window. Assume that the joint probability density function for \(X\) and \(Y\) is \(f(x, y)= \begin{cases}0.125 e^{-x / 4} e^{-y / 2} & \text { if } x \geq 0 \text { and } y \geq 0 \\ 0 & \text { otherwise }\end{cases}\) a. What is the probability that neither activity takes more than 5 minutes? b. What is the probability that you will complete your business at the bank (both activities) within 8 minutes?

INSURANCE SALES Let \(X\) be a random variable that measures the time (in minutes) that a person spends with an agent choosing a life insurance policy, and let \(Y\) measure the time (in minutes) the agent spends doing paperwork once the client has selected a policy. Suppose the joint probability density function for \(X\) and \(Y\) is \(f(x, y)= \begin{cases}\frac{1}{300} e^{-x / 30} e^{-y / 10} & \text { for } x \geq 0, y \geq 0 \\ 0 & \text { otherwise }\end{cases}\) a. Find the probability that choosing the policy takes more than 20 minutes. b. Find the probability that the entire transaction (policy selection and paperwork) will take more than half an hour. c. How much more time would you expect to spend selecting the policy than completing the paperwork?

TIME MANAGEMENT A shuttle tram arrives at a tram stop at a randomly selected time \(X\) within a 1-hour period, and a tourist independently arrives at the same stop also at a randomly selected time \(Y\) within the same hour. The tourist has the patience to wait for the tram for up to 20 minutes before calling a taxi. The joint probability density function for \(X\) and \(Y\) is $$ f(x, y)= \begin{cases}1 & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\ 0 & \text { otherwise }\end{cases} $$ a. What is the probability that the tram takes longer than 20 minutes to arrive? b. What is the probability that the tourist arrives after the tram? c. What is the probability that the tourist connects with the tram? [Hint: The event of this occurring has the form $$ Y+a \leq X \leq Y+b $$ for suitable numbers \(a\) and \(b\).]

WARRANTY PROTECTION A major appliance contains two components that are vital for its operation in the sense that if either fails, the appliance is rendered useless. Let the random variable \(X\) measure the useful life (in years) of the first component, and let \(Y\) measure the useful life of the second component (also in years). Suppose the joint probability density function for \(X\) and \(Y\) is \(f(x, y)= \begin{cases}0.1 e^{-x / 2} e^{-y / 5} & \text { if } x \geq 0 \text { and } y \geq 0 \\ 0 & \text { otherwise }\end{cases}\) a. Find the probability that the appliance fails within the first 5 years. b. Which component of a randomly selected appliance would you expect to last longer? How much longer?