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Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions

According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 5.8% tested positive for drugs. Of those testing positive, 22.5% were cocaine users and 54.4% marijuana users.1. What is the probability that of 10 workers testing positive, 2 are cocaine users, 5 are marijuana
As a student drives to school, he encounters a traffic signal.This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let X be the number of times he encounters a green light, X be
1. In Exercise 5.9, how many of the 15 trucks would you expect to have blowouts?2. What is the variance of the number of blowouts experienced by the 15 trucks? What does that mean?
If X represents the number of people in Exercise 5.13 who believe that antidepressants do not cure but only cover up the real problem, find the mean and variance of X when 10 people are selected at random.
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that 1. none contracts the disease;2. fewer than 2 contract the disease;3. more than 3 contract the disease.
The percentage of wins for the Chicago Bulls basketball team going into the playoffs for the 1996–97 season was 87.7.Round the 87.7 to 90 in order to use Table A.1.1. What is the probability that the Bulls sweep (4-0) the initial best-of-7 playoff series?2. What is the probability that the Bulls
A national study that examined attitudes about antidepressants revealed that approximately 70% of respondents believe “antidepressants do not really cure anything, they just cover up the real trouble.” According to this study, what is the probability that at least 3 of the next 5 people
In a survey of customers for a departmental store, it is reported that 80% of the customers are from higher economic group. What is the probability that fewer than 4 of the next 9 customers do not belong to higher income group?
The probability that a patient recovers from a delicate heart operation is 0.8. What is the probability that exactly 4 of the next 6 patients having this operation survive?
In a certain fitness test for athletes, it is found that 20% of the athletes fail to complete the test. Of the next 15 athletes tested, find the probability that 1. from 3 to 6 fail;2. fewer than 4 fail;3. more than 5 fail.
In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that 1. from 3 to 6 have blowouts;2. fewer than 4 have blowouts;3. more than 5 have blowouts.
According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state of Massachusetts first took Valium for psychological problems. Find the probability that among the next 8 users from this state who are interviewed, 1. exactly 3
One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct, 1. find the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers;2. find the probability that of 20 such patients recently admitted to
According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of vacation after they have been with the company for 15 years.Find the probability that among 8 companies surveyed at random, the number that give employees 4 weeks of vacation after
According to Chemical Engineering Progress (November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error.1. What is the probability that out of the next 20 pipework failures at least 10 are due to operator error?2. What is the probability that no more
In a certain city district, the need for money to buy drugs is stated as the reason for 75% of all thefts. Find the probability that among the next 5 theft cases reported in this district, 1. exactly 2 resulted from the need for money to buy drugs;2. at most 3 resulted from the need for money to
A student is selected from a class of 100 students to represent the class for a competitive event, by selecting a tag at random, from a box containing 100 tags numbered 1 to 100.Find the formula for the probability distribution of X representing the number on the tag drawn. What is the probability
Twelve people are given two identical speakers, which they are asked to listen to for differences, if any. Suppose that these people answer simply by guessing. Find the probability that three people claim to have heard a difference between the two speakers.
A random variable X that assumes the values x , x , …, x is called a discrete uniform random variable if its probability mass function is for all of x , x , …, x and 0 otherwise.Find the mean and variance of X.1 2 k 1 2 k
Project: Let X = number of hours each student in the class slept the night before. Create a discrete variable by using the following arbitrary intervals: X < 3, 3 ≤ X < 6, 6 ≤ X < 9, and X ≥ 9.1. Estimate the probability distribution for X.2. Calculate the estimated mean and variance for
Consider Review Exercise 3.73 on page 128. It involved Y, the proportion of impurities in a batch, and the density function is given by 12 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1. Find the expected percentage of impurities.2. Find the expected value of the proportion of quality material(i.e., find E(1 –
As we shall illustrate in Chapter 12, statistical methods associated with linear and nonlinear models are very important. In fact, exponential functions are often used in a wide variety of scientific and engineering problems. Consider a model that is fit to a set of data involving measured values k
Consider a ferry that can carry both buses and cars across a waterway. Each trip costs the owner approximately $10. The fee for cars is $3 and the fee for buses is $8. Let X and Y denote the number of buses and cars, respectively, carried on a given trip. The joint distribution of X and Y is given
A convenience store has two separate locations where customers can be checked out as they leave. These locations each have two cash registers and two employees who check out customers. Let X be the number of cash registers being used at 12 1 2 12 1 2 12 1 2 1a particular time for location 1 and Y
A delivery truck travels from point A to point B and back using the same route each day. There are four traffic lights on the route. Let X denote the number of red lights the truck encounters going from A to B and X denote the number encountered on the return trip. Data collected over a long period
It is known through data collection and considerable research that the amount of time in seconds that a certain employee of a company is late for work is a random variable X with density function In other words, he not only is slightly late at times, but also can be early to work.1. Find the
In business, it is important to plan and carry out research in order to anticipate what will occur at the end of the year.Research suggests that the profit (loss) spectrum for a certain company, with corresponding probabilities, is as follows:1. What is the expected profit?2. Give the standard
In a support system in the U.S. space program, a single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they all fail. Assume the components act
A company’s marketing and accounting departments have determined that if the company markets its newly developed product, the contribution of the product to the firm’s profit during the next 6 months will be described by the following:XY What is the company’s expected profit?
Consider Exercise 4.10 on page 137. Can it be said that the ratings given by the two experts are independent? Explain why or why not.
A dealer’s profit, in units of $5000, on a new automobile is a random variable X having density function 1. Find the variance of the dealer’s profit.2. Demonstrate that Chebyshev’s theorem holds for k = 2 with the density function above.3. What is the probability that the profit exceeds $500?
Consider random variables X and Y of Exercise 4.63 on page 158. Compute ρ .
Consider the joint density function Compute the correlation coefficient ρ .2 X Y XY
Consider the density function of Review Exercise 4.85.Demonstrate that Chebyshev’s theorem holds for k = 2 and k = 3.
Show that Cov(aX, bY) = ab Cov(X, Y).
Referring to the random variables whose joint density function is given in Exercise 3.40 on page 125, 1. find μ and μ ;2. find E[(X + Y)/2].
Suppose it is known that the life X of a particular compressor, in hours, has the density function 21. Find the mean life of the compressor.2. Find E(X ).3. Find the variance and standard deviation of the random variable X.
Referring to the random variables whose joint probability density function is given in Exercise 3.41 on page 125, find the expected weight for the sum of the creams and toffees if one purchased a box of these chocolates.
Referring to the random variables whose joint density function is given in Exercise 3.41 on page 125, find the covariance between the weight of the creams and the weight of the toffees in these boxes of chocolates.
Assume the length X, in minutes, of a particular type of telephone conversation is a random variable with probability density function 1. Determine the mean length E(X) of this type of telephone conversation.2. Find the variance and standard deviation of X.3. Find E[(X + 5) ].
Referring to the random variables whose joint probability density function is given in Exercise 3.47 on page 125, find the average amount of kerosene left in the tank at the end of the day.
Find the covariance of random variables X and Y having the joint probability density function
Prove Chebyshev’s theorem.
Compute P(μ – 2σ < X < μ + 2σ), where X has the density function and compare with the result given in Chebyshev’s theorem.Review Exercises
A random variable X has a mean μ = 10 and a variance σ =9. Using Chebyshev’s theorem, find YY 21. P(|X – 10| ≥ 9);2. P(|X – 10| < 9);3. P(4 < X < 16);4. the value of the constant c such that P(|X – 10| ≥c) ≤ 0.05.
120 new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 300 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns
An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on the package, has a mean life of 800 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 600 hours?Assume that the distribution is symmetric about the
Consider again the situation of Exercise 4.72. It is required to find Var(e ). Use Theorems 4.2 and 4.3 2Y YY Yand define Z = e . Thus, use the conditions of Exercise 4.73 to find Then do it not by using f(y), but rather by using the first-order Taylor series approximation to Var(e ). Comment!
For the situation in Exercise 4.72, compute E(e ) using Theorem 4.1, that is, by using Then compute E(e ) not by using f(y), but rather by using the second-order adjustment to the first-order approximation of E(e ). Comment.
A manufacturing company has developed a machine for cleaning carpet that is fuel-efficient because it delivers carpet cleaner so rapidly. Of interest is a random variable Y, the amount in gallons per minute delivered. It is known that the density function is given by 1. Sketch the density
The length of time Y, in minutes, required to generate a human reflex to tear gas has the density function 1. What is the mean time to reflex?2. Find E(Y2) and Var(Y).
Consider Review Exercise 3.64 on page 127. There are two service lines. The random variables X and Y are the proportions of time that line 1 and line 2 are in use, respectively. The joint probability density function for (X, Y) is given by 1. Determine whether or not X and Y are independent.2. It
Consider Review Exercise 3.77 on page 128. The random variables X and Y represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period in the day. The joint distribution is for x = 0, 1, 2, … and y = 0, 1, 2, ….1. Give E(X), E(Y), Var(X), and
The power P in watts which is dissipated in an electric circuit with resistance R is known to be given by P = I R, where I is current in amperes and R is a constant fixed at 50 ohms.However, I is a random variable with μ = 10 amperes and σ =0.04 amperes . Give numerical approximations to the mean
If the joint density function of X and Y is given by Find the expected value of .
Let X represent the number that occurs when a green die is tossed and Y the number that occurs when a red die is tossed.Find the variance of the random variable 1. 2X – Y;2. X + 4 Y – 50.
Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a green die is tossed.Find 1. E(X + Y);2. E(X – Y);3. E(4XY).
Suppose that X and Y are independent random variables with probability densities and 2 2 XY and Find the expected value of Z = XY.
Repeat Exercise 4.62 if X and Y are not independent andσ = 1.
If X and Y are independent random variables with variances and , find the variance of the random variable Z = – 2X + 4Y – 3.
Use Theorem 4.7 to evaluate E(2XY – X Y) for the joint probability distribution shown in Table 3.1 on page 116.
Suppose that X and Y are independent random variables having the joint probability distribution Find 1. E(3Y – 2X);2. E(YX).
If a random variable X is defined such that E[(X – 1) ] = 5 and E[(X – 2) ] = 2, find μ and σ .2 22 22 2
The total time, measured in units of 100 hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable X that has the density function Use Theorem 4.6 to evaluate the mean of the random variable Y= 60X + 48X, where Y is equal to the number of kilowatt hours
Let X be a random variable with the following probability distribution:Find E(X) and E(X ) and then, using these values, evaluate E[(2X+ 1) ].
Repeat Exercise 4.43 on page 147 by applying Theorem 4.5 and Corollary 4.6.
Using Theorem 4.5 and Corollary 4.6, find the mean and variance of the random variable Z = 5X + 3, where X has the probability distribution of Exercise 4.36 on page 147.
Referring to Exercise 4.35 on page 147, find the mean and variance of the discrete random variable Z = 3X – 2, when X represents the number of errors per 100 lines of code.
For the random variables X and Y in Exercise 3.39 on page 125, determine the correlation coefficient between X and Y.
For a laboratory assignment, if the equipment is working, the density function of the observed outcome X is Find the variance and standard deviation of X.
Consider the situation in Exercise 4.32 on page 139. The distribution of the number of imperfections per 10 meters of synthetic failure is given by Find the variance and standard deviation of the number of imperfections.
Given a random variable X, with standard deviation σ , and a random variable Y = a + bX, show that if b < 0, the correlation coefficient ρ = –1, and if b > 0, ρ = 1.X XY XY
For the random variables X and Y whose joint density function is given in Exercise 3.40 on page 125, find the covariance.
Find the covariance of the random variables X and Y of Exercise 3.44 on page 125.
Find the covariance of the random variables X and Y of Exercise 3.49 on page 126.
Find the covariance of the random variables X and Y of Exercise 3.39 on page 125.
Using the results of Exercise 4.21 on page 138, find the variance of g(X) = X , where X is a random variable having the density function given in Exercise 4.12 on page 137.2 2
Find the standard deviation of the random variable h(X) =(2X + 1) in Exercise 4.17 on page 118.
Referring to Exercise 4.14 on page 117, find for the function g(X) = 2 X + 4.
The total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a random variable X having the density function given in Exercise 4.13 on page 137. Find the variance of X.
The proportion of people who respond to a certain mailorder solicitation is a random variable X having the density function given in Exercise 4.14 on page 137. Find the variance of X.
A dealer’s profit, in units of $5000, on a new automobile is a random variable X having the density function given in Exercise 4.12 on page 137. Find the variance of X.
The random variable X, representing the number of errors per 100 lines of software code, has the following probability distribution:Using Theorem 4.2 on page 141, find the variance of X.
Use Definition 4.3 on page 140 to find the variance of the random variable X of Exercise 4.7 on page 137.
In Exercise 3.13 on page 112, the distribution of the number of imperfections per 10 meters of synthetic fabric is given by 1. plot the probability function.2. Find the expected number of imperfections, E(X) = μ.3. Find E(X ).4.2 Variance and Covariance of Random Variables The mean, or expected
Consider Exercise 3.32 on page 114.1. What is the mean proportion of the budget allocated to environmental and pollution control?2. What is the probability that a company selected at random will have allocated to environmental and pollution control a proportion that exceeds the population mean
In Exercise 3.31 on page 114, the distribution of times before a major repair of a washing machine was given as What is the population mean of the times to repair?
Exercise 3.29 on page 113 dealt with an important particle size distribution characterized by 1. plot the density function.2. Give the mean particle size.
Consider the information in Exercise 3.28 on page 113. The problem deals with the weight in ounces of the product in a cereal box, with 1. Plot the density function.2. Compute the expected value, or mean weight, in ounces.3. Are you surprised at your answer in (b)? Explain why or why not.
In Exercise 3.27 on page 113, a density function is given for the time to failure of an important component of a DVD player.Find the mean number of hours to failure of the component and thus the DVD player.
Let X and Y be random variables with joint density function Find the expected value of .
Referring to the random variables whose joint probability distribution is given in Exercise 3.51 on page 126, find the mean 2X Y 2X Y for the total number of jacks and kings when 3 cards are drawn without replacement from the 12 face cards of an ordinary deck of 52 playing cards.
Referring to the random variables whose joint probability distribution is given in Exercise 3.39 on page 125, 1. find E(X Y – 2XY);2. find μ – μ .
Suppose that X and Y have the following joint probability function f(x, y):1. Find the expected value of g(X, Y) = XY .2. Find μ and μ .
The hospitalization period, in days, for patients following treatment for a certain type of kidney disorder is a random 2–X 2variable Y = X + 3, where, X has the density function Find the average number of days that a person is hospitalized following treatment for this disorder.
What is the dealer’s average profit per automobile if the profit on each automobile is given by g(X) = X , where X is a random variable having the density function of Exercise 4.12?
A continuous random variable X has the density function Find the expected value of g(X) = e .
A large industrial firm purchases several new computers at the end of each year, the exact number depending on the frequency of repairs in the previous year. Suppose that the number of computers, X, purchased each year has the following probability distribution:If the cost of the desired model is
Find the expected value of the random variable g(X) = X , where X has the probability distribution of Exercise 4.2.X 1 2 g(X)2 2
Let X be a random variable with the following probability distribution:Find μ , where g(X) = (2X + 1) .

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