Prove Chebyshev’s theorem.
> Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. T
> An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the number of cars per year that will experience the catastroph
> Find the mean and variance of the random variable X in Exercise 5.61, representing the number of persons among 10,000 who make an error in preparing their income tax returns.
> Find the mean and variance of the random variable X in Exercise 5.58, representing the number of hurricanes per year to hit a certain area of the eastern United States.
> The probability that a student at a local high school fails the screening test for scoliosis (curvature of the spine) is known to be 0.004. Of the next 1875 students at the school who are screened for scoliosis, find the probability that (a) fewer than 5
> Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random and examined, find the probability that 6, 7, or 8 of them contain an error.
> The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. Find the probability that fewer than 7 field mice are found (a) on a given acre; (b) on 2 of the next 3 acres inspected.
> 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 6 companies surveyed at random, the number t
> Suppose the probability that any given person will believe a tale about the transgressions of a famous actress is 0.8. What is the probability that (a) the sixth person to hear this tale is the fourth one to believe it? (b) the third person to hear this
> A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the probability that in a given year that area will be hit by (a) fewer than 4 hurricanes; (b) anywhere from 6 to 8 hurricanes.
> On average, a textbook author makes two word processing errors per page on the first draft of her textbook. What is the probability that on the next page she will make (a) 4 or more errors? (b) no errors?
> On average, 3 traffic accidents per month occur at a certain intersection. What is the probability that in any given month at this intersection (a) exactly 5 accidents will occur? (b) fewer than 3 accidents will occur? (c) at least 2 accidents will occur
> The probability that a student pilot passes the written test for a private pilot’s license is 0.7. Find the probability that a given student will pass the test (a) on the third try; (b) before the fourth try.
> According to a study published by a group of University of Massachusetts sociologists, about two thirds of the 20 million persons in this country who take Valium are women. Assuming this figure to be a valid estimate, find the probability that on a given
> An inventory study determines that, on average, demands for a particular item at a warehouse are made 5 times per day. What is the probability that on a given day this item is requested (a) more than 5 times? (b) not at all?
> A scientist inoculates mice, one at a time, with a disease germ until he finds 2 that have contracted the disease. If the probability of contracting the disease is 1/6, what is the probability that 8 mice are required?
> Three people toss a fair coin and the odd one pays for coffee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 4 tosses are needed.
> Find the probability that a person flipping a coin gets (a) the third head on the seventh flip; (b) the first head on the fourth flip.
> According to Chemical Engineering Progress (November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error. (a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator
> The probability that a person living in a certain city owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in that city is the fifth one to own a dog.
> Every hour, 10,000 cans of soda are filled by a machine, among which 300 underfilled cans are produced. Each hour, a sample of 30 cans is randomly selected and the number of ounces of soda per can is checked. Denote by X the number of cans selected that
> A government task force suspects that some manufacturing companies are in violation of federal pollution regulations with regard to dumping a certain type of product. Twenty firms are under suspicion but not all can be inspected. Suppose that 3 of the fi
> A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system, a random sample of 5 is selected and all are tested. Suppose there are 2 faulty com
> Biologists doing studies in a particular environment often tag and release subjects in order to estimate 158 Chapter 5 Some Discrete Probability Distributions the size of a population or the prevalence of certain features in the population. Ten animals o
> An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample of 5 balls, find the probability that both blue balls and at least 1 red ball are selected.
> 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
> 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 a particular time
> 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 X1 denote the number of red lights the truck encounters going from A to B and X2 denote the number encountered on the
> 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, b
> 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: (a) Wha
> 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
> 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
> Consider Exercise 4.10 on page 117. Can it be said that the ratings given by the two experts are independent? Explain why or why not. Exercise 4.10: Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point s
> A dealer’s profit, in units of $5000, on a new automobile is a random variable X having density function (a) Find the variance of the dealer’s profit. (b) Demonstrate that Chebyshev’s theorem holds
> Consider random variables X and Y of Exercise 4.63 on page 138. Compute ρXY. Exercise 4.63: Repeat Exercise 4.62 if X and Y are not independent and σXY = 1. Exercise 4.62: If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, f
> A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is selected at random, find the probability that (a) all nationalities are represented; (b) all nationalities except Italian are represent
> A private pilot wishes to insure his airplane for $200,000. The insurance company estimates that a total loss will occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with probability 0.1. Ignoring all other partial losses, wha
> Consider the joint density function Compute the correlation coefficient ρXY. 16у , x > 2, 0 < y < 1, (0, f (x, y) = elsewhere.
> Consider the density function of Review Exercise 4.85. Demonstrate that Chebyshev’s theorem holds for k = 2 and k = 3. Review Exercise 4.85: Suppose it is known that the life X of a particular compressor, in hours, has the density func
> 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 105, (a) find μX and μY ; (b) find E[(X + Y )/2].
> Suppose it is known that the life X of a particular compressor, in hours, has the density function (a) Find the mean life of the compressor. (b) Find E(X2). (c) Find the variance and standard deviation of the random variable X. 1/900 e 900 x > 0, f
> Referring to the random variables whose joint probability density function is given in Exercise 3.41 on page 105, 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 105, 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 (a) Determine the mean length E(X) of this type of telephone conversation. (b) Find the variance and standard deviati
> Referring to the random variables whose joint probability density function is given in Exercise 3.47 on page 105, find the average amount of kerosene left in the tank at the end of the day.
> Find the probability of being dealt a bridge hand of 13 cards containing 5 spades, 2 hearts, 3 diamonds, and 3 clubs.
> Find the covariance of random variables X and Y having the joint probability density function Sx + y, 0< x < 1, 0 < y < 1, [0, f(x, y) elsewhere.
> Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.22, 0.36, 0.28, and 0.14, respectively, that she will be able to sell it for a profit of $250, sell it for a profit of $150, break even,
> Compute P(μ − 2σ and compare with the result given in Chebyshev’s theorem. J бх (1 — а), 0 <х <1, 0, f (x) = elsewhere,
> A random variable X has a mean μ = 10 and a variance σ2 = 4. Using Chebyshev’s theorem, find (a) P(|X − 10| ≥ 3); (b) P(|X − 10| < 3); (c) P(5
> Seventy new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathemat
> An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on the package, has a mean life of 900 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 700 hours? Assum
> Consider again the situation of Exercise 4.72. It is required to find Var(eY ). Use Theorems 4.2 and 4.3 and define Z = eY . Thus, use the conditions of Exercise 4.73 to find Var(Z) = E(Z2 ) − [E(Z)]2 .
> For the situation in Exercise 4.72, compute E(eY ) using Theorem 4.1, that is, by using Then compute E(eY ) not by using f(y), but rather by using the second-order adjustment to the first-order approximation of E(eY ). Comment. E(e*) = | e" f(y) dy
> 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 functio
> A nationwide survey of 17,000 college seniors by the University of Michigan revealed that almost 70% disapprove of daily pot smoking. If 18 of these seniors are selected at random and asked their opinion, what is the probability that more than 9 but fewe
> The length of time Y, in minutes, required to generate a human reflex to tear gas has the density function (a) What is the mean time to reflex? (b) Find E(Y2) and Var(Y ). Sie-/4, 0<y < , f(y) [ 0, elsewhere.
> Consider Review Exercise 3.64 on page 107. 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 (a) Determi
> By investing in a particular stock, a person can make a profit in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?
> Consider Review Exercise 3.77 on page 108. 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 (a) 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 = I2R, where I is current in amperes and R is a constant fixed at 50 ohms. However, I is a random variable with μI = 15 amperes and σ2I = 0.03 amp
> If the joint density function of X and Y is given by find the expected value of S{ 0 < x < 1, 1< y < 2, (x + 2y), f (x, y) = 10, elsewhere, g() = + X?Y. X, Y
> 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 (a) 2X − Y; (b) X + 3Y − 5.
> 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 (a) E(X + Y ); (b) E(X − Y ); (c) E(XY ).
> Suppose that X and Y are independent random variables with probability densities and and Find the expected value of Z = XY . x > 2, [0, g(x) elsewhere, 2y, 0<y < 1, h(y) 0, elsewhere.
> Repeat Exercise 4.62 if X and Y are not independent and σXY = 1. Exercise 4.62: If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, find the variance of the random variable Z = −2X + 4Y − 3.
> It is estimated that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at most 7 favor the new tax?
> 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, (a) exactly 2 resulted from the need for money to buy drugs; (b)
> If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, find the variance of the random variable Z = −2X + 4Y − 3.
> Use Theorem 4.7 to evaluate E(2XY2 − X2Y ) for the joint probability distribution shown in Table 3.1 on page 96.
> Suppose that X and Y are independent random variables having the joint probability distribution Find (a) E(2X − 3Y ); (b) E(XY ). f(x, y) 4 0.10 0.15 3 0.20 1 0.30 0.10 0.15
> An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4:00 P.M. and 5:00 P.M
> If a random variable X is defined such that E[(X − 1)2] = 10 and E[(X − 2)2 ]=6, find μ and σ2.
> 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 = 60X2 + 39X, wh
> Let X be a random variable with the following probability distribution: Find E(X) and E(X2) and then, using these values, evaluate E[(2X + 1)2]. -3 6 9 f (x) 6 3
> Repeat Exercise 4.43 on page 127 by applying Theorem 4.5 and Corollary 4.6. Exercise 4.43: The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y = 3X −2, where X has
> Suppose that a grocery store purchases 5 cartons of skim milk at the wholesale price of $1.20 per carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is removed from the shelf and the grocer receives a credit from
> 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 127. Exercise 4.36: Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectiv
> An annexation suit against a county subdivision of 1200 residences is being considered by a neighboring city. If the occupants of half the residences object to being annexed, what is the probability that in a random sample of 10 at least 3 favor the anne
> Referring to Exercise 4.35 on page 127, 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. Exercise 4.35: The random variable X, representing the number
> Random variables X and Y follow a joint distribution Determine the correlation coefficient between X and Y . (2, 0 <x <y < 1, 0, otherwise. f (x, y) =
> For the random variables X and Y in Exercise 3.39 on page 105, determine the correlation coefficient between X and Y . Exercise 3.39: From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. I
> 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. 2(1 — 2), 0 <х<1, (0, f (x) = otherwise.
> In a gambling game, a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?
> Consider the situation in Exercise 4.32 on page 119. 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. 1 3 4 f(x) 0.41 0.37 0.16 0
> Given a random variable X, with standard deviation σX, and a random variable Y = a + bX, show that if b < 0, the correlation coefficient ρXY = −1, and if b > 0, ρXY = 1.
> For the random variables X and Y whose joint density function is given in Exercise 3.40 on page 105, find the covariance.
> Find the covariance of the random variables X and Y of Exercise 3.44 on page 105.
> Find the covariance of the random variables X and Y of Exercise 3.49 on page 106.
> Among 150 IRS employees in a large city, only 30 are women. If 10 of the employees are chosen at random to provide free tax assistance for the residents of this city, use the binomial approximation to the hypergeometric distribution to find the probabili
> Find the covariance of the random variables X and Y of Exercise 3.39 on page 105.
> The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y = 3X −2, where X has the density function Find the mean and variance of the random variable Y . Sie-/4, x
> Using the results of Exercise 4.21 on page 118, find the variance of g(X) = X2, where X is a random variable having the density function given in Exercise 4.12 on page 117.
> Find the standard deviation of the random variable g(X) = (2X + 1)2 in Exercise 4.17 on page 118.
> Referring to Exercise 4.14 on page 117, find σ2(X) for the function g(X)=3X2 + 4.
> A coin is biased such that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.
> 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 117. Find the variance of X.
> The proportion of people who respond to a certain mail-order solicitation is a random variable X having the density function given in Exercise 4.14 on page 117. 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 117. Find the variance of X.
