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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
4.56 Repeat Exercise 4.43 on page 147 by applying Theorem 4.5 and Corollary 4.6.
4.55 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 the distributor equal to three-fourths
4.54 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.
4.53 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.
4.52 Random variables X and Y follow a joint distribution f(x, y) =2, 0 < x ≤ y < 1, 0, otherwise.Determine the correlation coefficient between X and
4.51 For the random variables X and Y in Exercise 3.39 on page 125, determine the correlation coefficient between X and Y .
4.50 For a laboratory assignment, if the equipment is working, the density function of the observed outcome X is f(x) =3x2 , 0 < x < 1, 0, elsewhere.Find the variance and standard deviation of X.
4.49 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 x 0 1 2 3 4 f(x) 0.41 0.37 0.16 0.05 0.01 Find the variance and standard deviation of the number of imperfections.
4.48 Given a random variable X, with standard deviationσX , and a random variable Y = a + bX, show that if b < 0, the correlation coefficient ρX Y = −1, and if b > 0, ρX Y = 1.
4.47 For the random variables X and Y whose joint density function is given in Exercise 3.40 on page 125, find the covariance.
4.46 Find the covariance of the random variables X and Y of Exercise 3.44 on page 125.
4.45 Find the covariance of the random variables X and Y of Exercise 3.49 on page 126.
4.44 Find the covariance of the random variables X and Y of Exercise 3.39 on page 125.
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 the density function f(x) =14 e−x/4, x>0 0, elsewhere.Find the mean and variance of the random variable Y .
4.42 Using the results of Exercise 4.21 on page 138, find the variance of g(X) = X2, where X is a random variable having the density function given in Exercise 4.12 on page 137.
4.41 Find the standard deviation of the random variable h(X) = (3X + 1)2 in Exercise 4.17 on page 138.
4.40 Referring to Exercise 4.14 on page 137, findσ2 g (X ) for the function g(X) = 3X + 4.
4.39 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.
4.38 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 137. Find the variance of X.
4.37 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.
4.36 Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2, or 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random variable X representing the number of power failures striking this subdivision.
4.35 The random variable X, representing the number of errors per 100 lines of software code, has the following probability distribution:x 2 3 4 5 6 f(x) 0.01 0.25 0.4 0.3 0.04 Using Theorem 4.2 on page 141, find the variance of X.
4.34 Let X be a random variable with the following probability distribution:x −2 3 5 f(x) 0.3 0.3 0.4 Find the standard deviation of X.
4.33 Use Definition 4.3 on page 140 to find the variance of the random variable X of Exercise 4.7 on page 137.
4.32 In Exercise 3.13 on page 112, the distribution of the number of imperfections per 10 meters of synthetic fabric is given by x 0 1 2 3 4 f(x) 0.41 0.37 0.16 0.05 0.01(a) Plot the probability function.(b) Find the expected number of imperfections, E(X) = μ.(c) Find E(X2 ).
4.31 Consider Exercise 3.32 on page 114.(a) What is the mean proportion of the budget allocated to environmental and pollution control?(b) What is the probability that a company selected at random will have allocated to environmental and pollution control a proportion that exceeds the population
4.30 In Exercise 3.31 on page 114, the distribution of times before a major repair of a washing machine was given as f(y) =14 e−y /4, y≥ 0, 0, elsewhere.What is the population mean of the times to repair?
4.29 Exercise 3.29 on page 113 dealt with an important particle size distribution characterized by f(x) =3x−4, x>1, 0, elsewhere.(a) Plot the density function.(b) Give the mean particle size.
4.28 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 f(x) =25, 23.75 ≤ x ≤ 26.25, 0, elsewhere.(a) Plot the density function.(b) Compute the expected value, or mean weight, in ounces.(c) Are you surprised
4.27 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.
4.26 Let X and Y be random variables with joint density function f(x, y) =4xy, 0 < x, y < 1, 0, elsewhere.Find the expected value of Z =√X2 + Y 2 .
4.25 Referring to the random variables whose joint probability distribution is given in Exercise 3.51 on page 126, find the mean 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.
4.24 Referring to the random variables whose joint probability distribution is given in Exercise 3.39 on page 125,(a) find E(X2Y − 2XY );(b) find μX − μY.
4.23 Suppose that X and Y have the following joint probability function f(x, y):x y 2 4 1 0.15 0.10 3 0.25 0.25 5 0.15 0.10(a) Find the expected value of g(X, Y ) = XY 2 .(b) Find μX and μY .
4.22 The hospitalization period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 5, where X has the density function f(x) =32(x+4)3, x>0, 0, elsewhere.Find the average number of days that a person is hospitalized following treatment for
4.21 What is the dealer’s average profit per automobile if the profit on each automobile is given by g(X) = X2, where X is a random variable having the density function of Exercise 4.12?
4.20 A continuous random variable X has the density function f(x) =12 e−x/2, x>0, 0, elsewhere.Find the expected value of g(X) = eX/4 .
4.19 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:x 0 1 2 3 f(x) 1/10 3/10 2/5 1/5
4.18 Find the expected value of the random variable g(X) = X2, where X has the probability distribution of Exercise 4.2.
4.17 Let X be a random variable with the following probability distribution:x −3 6 9 f(x) 1/6 1/2 1/3 Find μg (X ), where g(X) = (2X + 1)2 .
4.16 Suppose that you are inspecting a lot of 1000 light bulbs, among which 30 are defectives. You choose two light bulbs randomly from the lot without replacement.Let X1 =1, if the 1st light bulb is defective.0, otherwise.X2 =1, if the 2nd light bulb is defective.0, otherwise.Find the
4.15 Assume that two random variables (X, Y ) are uniformly distributed on a circle with radiusa. Then the joint probability density function is f(x, y) = 1π a2, x2 + y2 ≤ a2 , 0, otherwise.Find μX , the expected value of X.
4.14 Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function f(x) =2(x+2)5 , 0 < x < 1, 0, elsewhere.
4.13 The density function of the continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given in Exercise 3.7 on page 112 as f(x) =⎧⎨⎩x, 0 < x < 1, 2 − x, 1 ≤ x < 2, 0, elsewhere.Find the average
4.12 If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function f(x) =2(1 − x), 0 < x < 1, 0, elsewhere, find the average profit per automobile.
4.11 The density function of coded measurements of the pitch diameter of threads of a fitting is f(x) =1 ln(2)(1+x) , 0 < x < 1, 0, elsewhere.Find the expected value of X.
4.10 Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y .y f(x, y) 1 2 3 1 0.10 0.05 0.02 x 2 0.10
4.9 A private pilot wishes to insure an airplane for$200,000. The insurance company estimates that a total loss will occur with probability 0.001, a 50% loss with probability 0.01, and a 25% loss with probability 0.2. Ignoring all other partial losses, what premium should the insurance company
4.8 Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.2, 0.3, 0.4, and 0.1, respectively, that she will be able to sell it for a profit of $250, sell it for a profit of $150, break even, or sell it for a loss of $150.What is her
4.7 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?
4.6 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. on any sunny Friday. Find the
4.5 In a gambling game, a woman is paid $2 if she draws a jack or a queen and $4 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?
4.4 A coin is biased such that a head is twice as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.
4.3 Find the mean of the random variable T representing the total of the three coins in Exercise 3.25 on page 113.
4.2 The probability distribution of the discrete random variable, X, is f(x) =5 x1 4x 3 45−x, x= 0, 1, 2, 3, 4, 5.Find the mean of X.
4.1 The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given in Exercise 3.13 on page 112 as x 0 1 2 3 4 f(x) 0.41 0.37 0.16 0.05 0.01 Find the average number of imperfections per 10 meters of this fabric.
3.81 Project: Take 5 class periods to observe the shoe color of individuals in class. Assume the shoe color categories are red, white, black, brown, and other.Complete a frequency table for each color category.(a) Estimate and interpret the meaning of the probability distribution.(b) What is the
3.80 Consider a system of components in which there are 4 independent components, each of which possesses an operational probability of 0.9. The system does have a redundancy built in such that it does not fail if 3 out of the 4 components are operational. What is the probability that the total
3.79 Another type of system that is employed in engineering work is a group of parallel components or a parallel system. In this more conservative approach, the probability that the system operates is larger than the probability that any component operates. The system fails only when all components
3.78 The behavior of series of components plays a huge role in scientific and engineering reliability problems.The reliability of the entire system is certainly no better than that of the weakest component in the series. In a series system, the components operate independently of each other. In a
3.77 Consider the random variables X and Y that represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period.These street corners are fairly close together so it is important that traffic engineers deal with them jointly if necessary. The joint
3.76 Consider the situation of Review Exercise 3.75.But suppose the joint distribution of the two proportions is given by f(x1, x2) =6x2 , 0 < x2 < x1 < 1, 0, elsewhere.(a) Give the marginal distribution fX1 (x1 ) of the proportion X1 and verify that it is a valid density function.(b) What is the
3.75 A chemical system that results from a chemical reaction has two important components among others in a blend. The joint distribution describing the proportions X1 and X2 of these two components is given by f(x1, x2) =2, 0 < x1 < x2 < 1, 0, elsewhere.(a) Give the marginal distribution of X1
3.74 The time Z in minutes between calls to an electrical supply system has the probability density function f(z) = 1 10 e−z /10 , 0 < z < ∞, 0, elsewhere.(a) What is the probability that there are no calls within a 20-minute time interval?(b) What is the probability that the first call comes
3.73 Impurities in a batch of final product of a chemical process often reflect a serious problem. From considerable plant data gathered, it is known that the proportion Y of impurities in a batch has a density function given by f(y) =10(1 − y)9 , 0 ≤ y ≤ 1, 0, elsewhere.(a) Verify that the
3.72 Passenger congestion is a service problem in airports.Trains are installed within the airport to reduce the congestion. With the use of the train, the time X, in minutes, that it takes to travel from the main terminal to a particular concourse has density function f(x) = 1 20 , 0 < x < 20 0,
3.71 The shelf life of a product is a random variable that is related to consumer acceptance. It turns out that the shelf life Y in days of a certain type of bakery product has a density function f(y) =13 e−y /3, y>0 0, elsewhere.What fraction of the loaves of this product stocked today would
3.70 Pairs of pants are being produced by a particular outlet facility. The pants are checked by a group of 10 workers. The workers inspect pairs of pants taken randomly from the production line. Each inspector is assigned a number from 1 through 10. A buyer selects a pair of pants for purchase.
3.69 The life span, in hours, of an electrical component is a random variable with cumulative distribution function F(x) =1 − e−x/75, x>0, 0, eleswhere.(a) Determine its probability density function.(b) Determine the probability that the life span of such a component will exceed 70 hours.
3.68 Consider the following joint probability density function of the random variables X and Y :f(x, y) =3x − y 11 , 1 < x < 3, 0 < y < 1, 0, elsewhere.(a) Find the marginal density functions of X and Y .(b) Are X and Y independent?(c) Find P(X > 2).
3.67 An industrial process manufactures items that can be classified as either defective or not defective.The probability that an item is defective is 0.2. An experiment is conducted in which 6 items are drawn randomly from the process. Let the random variable X be the number of defectives in this
3.66 Consider the random variables X and Y with joint density function f(x, y) =x + y, 0 ≤ x ≤ 1; 0 ≤ y ≤ 1 0, elsewhere.(a) Find the marginal distributions of X and Y .(b) Find P(X > 0.25, Y > 0.5).
3.65 Let the number of phone calls received by a switchboard during a 5-minute interval be a random variable X with probability function f(x) = e−2 2x x! , for x = 0, 1, 2, . . . .(a) Determine the probability that X equals 0, 1, 2, 3, 4, 5, and 6.(b) Graph the probability mass function for these
3.64 A service facility operates with two service lines.On a randomly selected day, let X be the proportion of time that the first line is in use whereas Y is the proportion of time that the second line is in use. Suppose that the joint probability density function for (X, Y ) is f(x, y) =32(x2 +
3.63 Two electronic components of a missile system work in harmony for the success of the total system.Let X and Y denote the life in hours of the two components.The joint density of X and Y is f(x, y) =ye−y (1+x), x,y≥ 0, 0, elsewhere.(a) Give the marginal density functions for both random
3.62 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 cumulative distribution function of X is F(x) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0, if x < 1,
3.61 A tobacco company produces blends of tobacco, with each blend containing various proportions of Turkish, domestic, and other tobaccos. The proportions of Turkish and domestic in a blend are random variables with joint density function (X = Turkish and Y = domestic)f(x, y) =24xy, 0 ≤ x, y
3.60 The joint probability density function of the random variables X, Y , and Z is f(x, y, z) =49 xyz2 , 0 < x,y < 1, 0 < z < 3, 0, elsewhere.Find(a) the joint marginal density function of X and Y ;(b) the marginal density of Z;(c) P(14< X < 12, Y > 13, 2 < Z < 3);(d) P(0 < Z < 2 | X = 12, Y =
3.59 Determine whether the two random variables of Exercise 3.44 are dependent or independent.
3.58 Determine whether the two random variables of Exercise 3.43 are dependent or independent.
3.57 Let X, Y , and Z have the joint probability density function f(x, y, z) =kxy2 z, 0 < x,y < 1, 0 < z < 2, 0, elsewhere.(a) Find k.(b) Find P(X < 14, Y > 12, 1 < Z < 2).
3.56 The joint density function of the random variables X and Y is f(x, y) =6x, 0 < x < 1, 0 < y < 1 − x, 0, elsewhere.(a) Show that X and Y are not independent.(b) Find P(X > 0.3 | Y = 0.5).
3.55 Determine whether the two random variables of Exercise 3.50 are dependent or independent.
3.54 Determine whether the two random variables of Exercise 3.49 are dependent or independent.
3.53 Given the joint density function f(x, y) =6−x−y 8 , 0 < x < 2, 2 < y < 4, 0, elsewhere, find P(1 < Y < 3 | X = 1.5).
3.52 A coin is tossed twice. Let Z denote the number of heads on the first toss and W, the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 30% chance of occurring, find(a) the joint probability distribution of W and Z;(b) the marginal distribution of W;(c) the
3.51 Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings) of an ordinary deck of 52 playing cards. Let X be the number of kings selected and Y the number of jacks.Find(a) the joint probability distribution of X and Y ;(b) P[(X, Y ) ∈ A], where A is the
3.50 Suppose that X and Y have the following joint probability distribution:x f(x, y) 2 4 1 0.10 0.15 y 3 0.20 0.30 5 0.10 0.15(a) Find the marginal distribution of X.(b) Find the marginal distribution of Y .
3.49 Let X denote the number of times a certain numerical control machine will malfunction: 1, 2, or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as xf(x, y) 1 2 3 y135 0.05 0.05 0.00 0.05 0.10
3.48 Referring to Exercise 3.39, find(a) f(y|2) for all values of y;(b) P(Y = 0 | X = 2).
3.47 The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x ≤ y, and assume that the joint density function of these variables
3.46 Referring to Exercise 3.38, find(a) the marginal distribution of X;(b) the marginal distribution of Y .
3.45 Let X denote the diameter of an armored electric cable and Y , the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density f(x, y) =1/y, 0 < x < y < 1, 0, elsewhere.Find P(X +Y > 2/3).
3.44 Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pounds per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density
3.43 Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F)at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density f(x, y) =4xy, 0 < x < 1, 0 < y < 1, 0, elsewhere.Find(a) P(0 ≤ X ≤ 12 and
3.42 Let X and Y denote the lengths of life, in years, of two components in an electronic system. If the joint density function of these variables is f(x, y) =e−(x + y ), x>0, y >0, 0, elsewhere,find P(1 < X < 2 | Y = 2).
3.41 A candy company distributes boxes of chocolates with a mixture of creams, toffees, and cordials.Suppose that the weight of each box is 1 kilogram, but the individual weights of the creams, toffees, and cordials vary from box to box. For a randomly selected box, let X and Y represent the
3.40 A fast-food restaurant operates both a drivethrough facility and a walk-in facility. On a randomly selected day, let X and Y , respectively, be the proportions of the time that the drive-through and walk-in facilities are in use, and suppose that the joint density function of these random
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. If X is the number of oranges and Y is the number of apples in the sample, find(a) the joint probability distribution of X and Y ;(b) P[(X, Y ) ∈ A], where A is the region
3.38 If the joint probability distribution of X and Y is given by f(x, y) = x + y 30 , for x = 0, 1, 2, 3; y = 0, 1, 2;find(a) P(X ≤ 1, Y = 1);(b) P(X > 1, Y ≤ 1);(c) P(X ≤ Y );(d) P(X + Y = 2).
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