For married couples living in a certain suburb, the probability that the husband will vote on a bond referendum is 0.21, the probability that the wife will vote on the referendum is 0.28, and the probability that both the husband and the wife will vote is 0.15. What is the probability that (a) at least one member of a married couple will vote? (b) a wife will vote, given that her husband will vote? (c) a husband will vote, given that his wife will not vote?
> In Exercise 3.13 on page 92, the distribution of the number of imperfections per 10 meters of synthetic fabric is given by (a) Plot the probability function. (b) Find the expected number of imperfections, E(X) = μ. (c) Find E(X2). 1 2
> Consider Exercise 3.32 on page 94. (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
> In Exercise 3.31 on page 94, 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? Sie-v/4, y 2 0, f(y) : (0, elsewhere.
> Find the mean of the random variable T representing the total of the three coins in Exercise 3.25 on page 93.
> Exercise 3.29 on page 93 dealt with an important particle size distribution characterized by (a) Plot the density function. (b) Give the mean particle size. -4 f(x) = S3x¯ x > 1, elsewhere.
> Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces of the product in a cereal box, with (a) Plot the density function. (b) Compute the expected value, or mean weight, in ounces. (c) Are you surprised at yo
> In Exercise 3.27 on page 93, 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.
> A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are fo
> Referring to the random variables whose joint probability distribution is given in Exercise 3.51 on page 106, 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 5
> Referring to the random variables whose joint probability distribution is given in Exercise 3.39 on page 105, (a) find E(X2Y − 2XY ); (b) find μX − μY .
> Suppose that X and Y have the following joint probability function: (a) Find the expected value of g(X, Y ) = XY2. (b) Find μX and μY . f(x, y) 4 1 0.10 0.15 3 0.20 0.30 0.10 0.15
> The hospitalization period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function Find the average number of days that a person is hospitalized following treatme
> 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?
> A continuous random variable X has the density function Find the expected value of g(X) = e2X/3. x > 0, 0, f (x) : elsewhere.
> The probability distribution of the discrete random variable X is Find the mean of X. 3-r 3 3 f (x) : x = 0,1,2, 3. 4
> A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous year. Suppose that the number of word processors, X, purchased each year has the following proba
> Find the expected value of the random variable g(X) = X2, where X has the probability distribution of Exercise 4.2.
> A company is interested in evaluating its current inspection procedure for shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of shipments with 20% de
> Let X be a random variable with the following probability distribution: Find μg(X), where g(X) = (2X + 1)2. -3 6 9 f (x) 1/6 1/2 1/3
> Suppose that you are inspecting a lot of 1000 light bulbs, among which 20 are defectives. You choose two light bulbs randomly from the lot without replacement. Let Find the probability that at least one light bulb chosen is defective. 1, if the 1st
> Assume that two random variables (X, Y) are uniformly distributed on a circle with radius a. Then the joint probability density function is Find μX, the expected value of X. S , a? +y? < a?, f(x, y) 10, otherwise.
> Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function 2(x+2) 0 < x < 1, f (x) = 0, elsewhere.
> 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 92 as Find the average number of hours per year
> 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 find the average profit per automobile. S 2(1 – x), 0<< x < 1, (0, f(x) = elsewhere,
> The density function of coded measurements of the pitch diameter of threads of a fitting is Find the expected value of X. 4 0 < x < 1, f (x) = T(1+x²)' 0, elsewhere.
> Consider Review Exercise 3.73 on page 108. It involved Y, the proportion of impurities in a batch, and the density function is given by (a) Find the expected percentage of impurities. (b) Find the expected value of the proportion of quality material (i
> 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
> 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. Fi
> What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 among 9 students, 4 of whom are minors?
> 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 92 as x 01234 Find the average number of imperfections per 10 meters of this fabr
> Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom
> If the person in Exercise 2.96 received a speeding ticket on her way to work, what is the probability that she passed through the radar trap located at L2? Exercise 2.96: Police plan to enforce speed limits by using radar traps at four different locatio
> Referring to Exercise 2.95, what is the probability that a person diagnosed as having cancer actually has the disease? Exercise 2.95: In a certain region of the country it is known from past experience that the probability of selecting an adult over 40
> Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations L1, L2, L3, and L4 will be operated 40%, 30%, 20%, and 30% of the time. If a person who is speeding on h
> In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.7
> In the situation of Exercise 2.93, it is known that the system does not work. What is the probability that the component A also does not work?
> A circuit system is given in Figure 2.11. Assume the components fail independently. (a) What is the probability that the entire system works? (b) Given that the system works, what is the probability that the component A is not working? Figure 2.11:
> Suppose the diagram of an electrical system is as given in Figure 2.10. What is the probability that the system works? Assume the components fail independently. Figure 2.10: 0.7 B 0.95 0.9 A D 0.8 Figure 2.10: Diagram for Exercise 2.92.
> Find the probability of randomly selecting 4 good quarts of milk in succession from a cooler containing 20 quarts of which 5 have spoiled, by using (a) the first formula of Theorem 2.12 on page 68; (b) the formulas of Theorem 2.6 and Rule 2.3 on pages 50
> If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that (a) exactly 2 of them will be face cards? (b) at least 1 of them will be a queen?
> Pollution of the rivers in the United States has been a problem for many years. Consider the following events: A: the river is polluted, B: a sample of water tested detects pollution, C: fishing is permitted. Assume P(A) = 0.3, P(B|A) = 0.75, P(B|A’) = 0
> The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function (a) Show that P(0 (b) Find the probability that more than 1/4 but fewer than 1/2 of the people contacted will resp
> A town has two fire engines operating independently. The probability that a specific engine is available when needed is 0.96. (a) What is the probability that neither is available when needed? (b) What is the probability that a fire engine is available w
> Before the distribution of certain statistical software, every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rates for the four testing programs ar
> A real estate agent has 8 master keys to open several new homes. Only 1 master key will open any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate agent can get into a specific home if the agent se
> In 1970, 11% of Americans completed four years of college; 43% of them were women. In 1990, 22% of Americans completed four years of college; 53% of them were women (Time, Jan. 19, 1996). (a) Given that a person completed four years of college in 1970, w
> The probability that a doctor correctly diagnoses a particular illness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnos
> The probability that the head of a household is home when a telemarketing representative calls is 0.4. Given that the head of the house is home, the probability that goods will be bought from the company is 0.3. Find the probability that the head of the
> The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.12; the probability that it is a camper is 0.28; and the probability that it is a camper with Canadian license plates is 0.09. What is the probability that (a) a c
> From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective missiles that will not fire, what is the probability that (a) all 4 will fire? (b) at most 2 will not fire?
> The probability that a married man watches a certain television show is 0.4, and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that his wife does, is 0.7. Find the probability that (a) a
> Consider a system of components in which there are 5 independent components, each of which possesses an operational probability of 0.92. The system does have a redundancy built in such that it does not fail if 3 out of the 5 components are operational. W
> Find the probability distribution of the random variable W in Exercise 3.3, assuming that the coin is biased so that a head is twice as likely to occur as a tail. Exercise 3.3: Let W be a random variable giving the number of heads minus the number of ta
> 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. T
> 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 op
> 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 th
> Consider the situation of Review Exercise 3.75. But suppose the joint distribution of the two proportions is given by (a) Give the marginal distribution fX1 (x1) of the proportion X1 and verify that it is a valid density function. (b) What is the proba
> 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 (a) Give the marginal distribution of X1. (b)
> The time Z in minutes between calls to an electrical supply system has the probability density function (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 within 1
> 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 (a) Verify that the above is
> A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2 ≤ X ≤ 3).
> 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
> 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 What fraction of the loaves of this product stocked today wou
> 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
> The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function Find the probability that over a period of one year, a family runs
> The life span in hours of an electrical component is a random variable with cumulative distribution function (a) Determine its probability density function. (b) Determine the probability that the life span of such a component will exceed 70 hours.
> Consider the following joint probability density function of the random variables X and Y: (a) Find the marginal density functions of X and Y. (b) Are X and Y independent? (c) Find P(X > 2). 3*-, 1<x < 3, 1< y < 2, 0, f(x, y) = elsewhere.
> An industrial process manufactures items that can be classified as either defective or not defective. The probability that an item is defective is 0.1. An experiment is conducted in which 5 items are drawn randomly from the process. Let the random variab
> Consider the random variables X and Y with joint density function (a) Find the marginal distributions of X and Y . (b) Find P(X > 0.5,Y > 0.5). Sr +y, 0<x, y <1, 10, f(x, y) = elsewhere.
> Let the number of phone calls received by a switchboard during a 5-minute interval be a random variable X with probability function (a) Determine the probability that X equals 0, 1, 2, 3, 4, 5, and 6. (b) Graph the probability mass function for these v
> 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 f
> To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin tablets that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the travele
> An employee is selected from a staff of 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to 10. Find the formula for the probability distribution of X representing the number on the tag that is
> 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 (a) Give the marginal density functions for both random varia
> 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 (a) What is the p
> 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
> The joint probability density function of the random variables X, Y, and Z is Find (a) the joint marginal density function of Y and Z; (b) the marginal density of Y; (c) P( 1 4 1 3 , 1 (d) P(0 4xyz? , 0< x, Y < 1, 0 < z < 3, f(x, y, z) = 0, elsewh
> The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function Find the probability that a bottle of this medicine will have a shell life of (a) at least 200 days; (b) anywhere from 80 to 120 days
> Determine whether the two random variables of Exercise 3.44 are dependent or independent. Exercise 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 pr
> Determine whether the two random variables of Exercise 3.43 are dependent or independent. Exercise 3.43: Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (â—¦F) at which a certain reaction
> Let X, Y, and Z have the joint probability density function (a) Find k. (b) Find P(X 1 2 , 1 S kxy°z, 0< x, y < 1, 0 < z < 2, 10, f(x, y, z) = elsewhere.
> The joint density function of the random variables X and Y is (a) Show that X and Y are not independent. (b) Find P(X > 0.3 | Y = 0.5). f (x, y) S6x, 0< x < 1, 0 < y <1 – x, 1o, elsewhere.
> Determine whether the two random variables of Exercise 3.50 are dependent or independent. Exercise 3.50: Suppose that X and Y have the following joint probability distribution: f(x, y) 2 4 1 0.10 0.15 0.20 0.30 0.10 0.15
> A homeowner plants 6 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 2 daffodil bulbs and 4 tulip bulbs?
> Determine whether the two random variables of Exercise 3.49 are dependent or independent. Exercise 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
> Given the joint density function find P(1 ーエー2,0<z< 2, 2<y<4, f(x, y) = elsewhere,
> 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 40% chance of occurring, find (a) the joint probability distribution of W and Z; (b) th
> 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;
> Suppose that X and Y have the following joint probability distribution: (a) Find the marginal distribution of X. (b) Find the marginal distribution of Y. f(x, y) 2 4 1 0.10 0.15 0.20 0.30 0.10 0.15
> 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 (a)
> Referring to Exercise 3.39, find (a) f(y|2) for all values of y; (b) P(Y = 0 | X = 2). 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. If X is the number of oranges and
> 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
> A manufacturer knows that on average 20% of the electric toasters produced require repairs within 1 year after they are sold. When 20 toasters are randomly selected, find appropriate numbers x and y such that (a) the probability that at least x of them w
> Let X denote the diameter of an armored electric cable and Y denote 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 Find P(X + Y > 1/2).
> 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
> Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (â—¦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density Find (a) P(0 &aci
> 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 find P(0 Se-(z+y), x> 0, y > 0, 10, т > 0, у > 0, elsewhere, f(x, y) :
> 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 selec
> A fast-food restaurant operates both a drive through 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 joi
> A coin is flipped until 3 heads in succession occur. List only those elements of the sample space that require 6 or less tosses. Is this a discrete sample space? Explain.
