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Probability & Statistics For Engineers & Scientists 7th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying - Solutions

Let X denotes the diameter of an armored electric cable and V denotes the diameter of the ceramic mold that makes the cable. Both X and Y is scaled so that they range between 0 and 1. Suppose that X and Y have the joint density Find P(X + Y > 1/2).
Referring to Exercise 3.38, find(a) The marginal distribution of X;(b) The marginal distribution of Y.
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 is(a)
Referring to Exercise 3.39 find(a) f(y\2) for all values of y;(b) P(Y = 0 | X = 2).
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) Evaluate the marginal distribution of X.(b)
Suppose that X and Y have the following joint probability distribution:(a) Find the marginal distribution of A.(b) Find the marginal distribution of Y.
Consider an experiment that consists of 2 rolls of a balanced die. If X is the number of 4s and Y is the number of 5s obtained in the 2 rolls of the die, find (a) The joint probability distribution of A and V; (b) P [(X, Y) € A], where A is the region {(x, y) 2x + y < 3}.
Let X denote the: number of heads and Y the number of heads minus the number of tails when 3 coins are tossed. Find the joint probability distribution of X and Y.
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) e A) where A is the region given
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) The marginal distribution of W;(c) The marginal
Given the joint density function find P (l < Y < 3 | X = 1).
Determine whether the two random variables of Exercise 3.49 are dependent or independent.
Determine whether the two random variables of Exercise 3.50 are dependent or independent.
The joint density function of the random variables X and Y is(a) Show that X and V are not independent.(b) Find P(X > 0.3 | Y = 0.5).
Let X, Y, and Z have the joint probability density function(a) Find k.(b) Find P (X < 1/4, y > ½ , l < Z < 2 ) .
Determine whether the two random variables of Exercise 3.43 are dependent or independent.
Determine whether the two random variables of Exercise 3.44 are dependent or independent.
The joint probability density function of the random variables X, Y, and Z is find(a) The joint marginal density functions of Y and Z:(b) The marginal density of Y;(c) P (1/4 < X < 2 ½, Y > ½, KZ < 2);(d) P (0 < X < ½ | Y = i 2 = 2).
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)(a) Find the probability that in a
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 probability mass function of X?(b) Compute P(4 <
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 variables.(b) What is the probability that both
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 (A, V) is(a) Compute the
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 values of x.(c) Determine the cumulative
Consider the random variables X and Y with joint density function(a) Find the marginal distributions of X and Y.(b) Find P(A > 0 . 5 , y >0.5).
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 variable X be the number of defectives in this
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).
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.
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. Let
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 would you expect to be sellable 3 days from now?
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 that it takes in minutes to travel from the main terminal to a particular concourse has density function(a) Show that the pdf above is a
Impurities in the 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 valid density function.(b) A batch is
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 10 minutes of opening?
A chemical system that results from a chemical reaction has two important components among others in a blend. The joint distribution describing the proportion A1 and X2 of these two components is given by(a) Give the marginal distribution of X1.(b) Give the marginal distribution of X2.(c) What is
Consider the situation of Review Exercise 3.77. But suppose the joint distribution of the two proportions is given by(a) Give the marginal distribution fX1, (xi) of the proportion Xi and verify that it is a valid density function.(b) What is the probability that proportion X2 is less than 0.5 given
Consider the random variables X and Y that represent the number of vehicles that arrive at 2 separate street corners during a certain 2-tninute period. These street corners arc fairly close together so it is important that traffic engineers deal with them jointly if necessary. The joint
The behavior of series of components play a huge role in scientific and engineering reliability problems. The reliability of the entire system is certainly no better than the weakest component in the series. In a series system, the components operate independently of each other. In a particular
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 systems fail.
Consider a system of components in which there are five 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. What is the probability that the total
List the elements of each of the following sample spaces:(a) The set of integers between 1 and 50 divisible by 8:(b) The set S = {x | x2 + 4x – 5 = 0};(c) The set of outcomes when a coin is tossed until a tail or three heads appear:(d) The set S = {x | a; is a continent};(e) The set. S = {x \ 2x
List the elements of each of the following sample spaces:(a) The set of integers between 1 and 50 divisible by 8:(b) The set S = {x | x2 + 4x – 5 = 0};(c) The set of outcomes when a coin is tossed until a tail or three heads appear:(d) The set S = {x | a; is a continent};(e) The set. S = {x \ 2x
Use the rule method to describe the sample space S consisting of all points in the first quadrant inside a circle of radius 3 with center at the origin.
Which of the following events are equal?(a) A = {1, 3};(b) B = {x | x is a number on a die}:(c) C = {xx2 – 4x + 3 = 0};(d) D = {x | x is the number of heads when six coins are tossed}.
An experiment involves tossing a pair of dice, 1 green and 1 red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S(a) By listing the elements {x, y);(b) By using the rule method.
An experiment consists of tossing a die and then flipping a coin once if the number on the: die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote
Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A1 A3, for example, to denote the simple event that alternates 1 and 3 are selected, list the 0 elements of the sample space S.
Four students are selected at random from a chemistry class and classified as male or female. List the elements of the sample space Si using the letter M for "male" and F for "female." Define a second sample space S2 where the elements represent the number of females selected.
For the sample space of Exercise 2.4:(a) List the elements corresponding to the event A that the sum is greater than 8;(b) List the elements corresponding to the event B that a 2 occurs on either die;(c) List the elements corresponding to the event C that a number greater than 4 comes up on the
For the sample space of Exercise 2.5,(a) List the elements corresponding to the event A that a number less than 3 occurs on the die;(b) List the elements corresponding to the event B that 2 tails occur;(c) List the elements corresponding to the event A’:(d) List the elements corresponding to the
An engineering firm is hired to determine if certain waterways in Virginia are safe for fishing. Samples are taken from three rivers.(a) List the elements of a sample space S, using the letters F for "safe to fish" and N for "not safe to fish."(b) List the elements of S corresponding to event E
The resumes of 2 male applicants for a college teaching position in chemistry are placed in the same file as the resumes of 2 female applicants. Two positions become available and the first, at the rank of assistant professor, is filled by selecting 1 of the 4 applicants at random. The second
Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of subjects will be used to study the effect of exercise. Group one is sedentary while group two walks and group three swims for 1 hour a day. Half of each of the three exercise groups
Construct a Venn diagram to illustrate the possible intersections and unions for the following events relative to the sample space consisting of all automobiles made in the United States. F: Four door, S: Sun roof, P: Power steering.
If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, and D = {1, 6, 7}, list the elements of the sets corresponding to the following events: (a) A U C; (b) A ∩ B; (c) C'; (d) (C' ∩ D) U B; (e) (S∩C)'; (f) A ∩ C n D'.
Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc}, and the eventsA = {copper, sodium, zinc},B = {sodium, nitrogen, potassium},C = {oxygen}.List the elements of the sets corresponding to the following events:(a) A';(b) AUC;(c) (A ∩ B') U C;(d) B' n C";(e) A
If S = {x | 0 < x < 12}, M = {x | 1 < x < 9}, and N = {x | 0 < x < 5}, find(a) M U N;(b) M n N;(c) M' n N'.
Let A. B, and C be events relative to the sample space S. Using Venn diagrams, shade the areas representing the following events: (a) (A ∩ B)’; (b) (A U B)' ; (c) (.4 n C) U B.
Which of the following pairs of events are mutually exclusive?(a) A golfer scoring the lowest 18-hole round in a 72-hole tournament and losing the tournament.(b) A poker player getting a flush (all cards in the same suit) and 3 of a kind on the same 5-card hand.(c) A mother giving birth to a baby
Suppose that a family is leaving on a summer vacation in their camper and that M is the event that they will experience mechanical problems, T is the event that they will receive a ticket; for committing a traffic violation and V is the event: that they will arrive at a campsite with no vacancies.
Referring to Exercise 2.19 and the Venn diagram of Figure list the numbers of the regions that represent, the following events:(a) The family will experience no mechanical problems and commit no traffic violation but will arrive at a campsite with no vacancies.(b) The family will experience both
Registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many ways can a person arrange to go on a sightseeing tour planned by this convention?
In a medical study patients are classified in 8 ways according to whether they have blood type AB+, AB–, A+, A–, B+, B–, O+, or O–, and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classified.
If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are there in the sample space?
Students at a private liberal arts college are classified as being freshmen, sophomores, juniors, or seniors, and also according to whether they are male or female. Find the total number of possible classifications for the students of that college.
A certain shoe conies in 5 different styles with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many different pairs would the store have on display?
A California study concluded that by following 7 simple health rules a man's life can be extended by 11 years on the average and a woman's life by 7 years. These 7 rules are as follows: no smoking, regular exercise, use alcohol moderately, get, 7 to 8 hours of sleep, maintain proper weight, eat,
A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different, plans are available to this buyer?
A drug for the relief of asthma can be purchased from 5 different, manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma?
In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed?
In how many different ways can a true-false test consisting of 9 questions be answered?
If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct,(a) In how many different ways can a student check off one answer to each question?(b) In how many ways can a student, check off one answer to each question and get all the answers wrong?
(a) How many distinct permutations can be made from the letters of the word columns?(b) How many of these permutations start with the letter mi;
A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 3 digits, the first of which is a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are different, find the maximum number of automobile registrations
(a) In how many ways can 6 people be lined up to get on a bus?(b) If 3 specific persons, among 6, insist on following each other, how many ways are possible?(c) If 2 specific persons, among 6, refuse to follow each other, how many ways are possible?
A contractor wishes to build 9 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side of the street and 3 lots are on the opposite side?
(a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6, if each digit can be used only once? (b) How many of these are odd numbers?(c) How many are greater than 330?
In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate?
Four married couples have bought 8 scats in the same row for a concert. In how many different ways can they be seated?(a) With no restrictions?(b) If each couple is to sit together?(c) If all the men sit together to the right of all the women?
In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space S for the number of possible orders at the conclusion of the contest for(a) All 8 finalists;(b) The first 3 positions.
In how many ways can 5 starting positions on a basketball team be filled with 8 men who can play any of the positions?
Find the number of ways that 6 teachers can be assigned to 4 sections of an introductory psychology course if no teacher is assigned to more than one section.
Three lottery tickets for first, second, and third prizes are drawn from a group of 40 tickets. Find the number of sample points in S for awarding the 3 prizes if each contestant holds only 1 ticket.
In how many ways can 5 different trees be planted in a circle?
In how many ways can a caravan of 8 covered wagons from Arizona be arranged in a circle?
How many distinct permutations can be made from the letters of the word infinity?
In how many ways can 3 oaks, 4 pines, and 2 maples be arranged along a property line if one does not distinguish among trees of the same kind?
A college plays 12 football games during a season. In how many ways can the team end the season with 7 wins, 3 losses, and 2 ties?
Nine people are going on a skiing trip in 3 cars that hold 2, 4, and 5 passengers, respectively. In how many ways is it possible to transport the 9 people to the ski lodge, using all cars?
How many ways are there to select 3 candidates from 8 equally qualified recent graduates for openings in an accounting firm?
How many ways are there that no two students will have the same birth date in a size of 60 classes?
Find the errors in each of the following statements:(a) The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given day in February are, respectively, 0.19, 0.38, 0.29, and 0.15.(b) The probability that it will rain tomorrow is 0.40 and the probability that it will
Assuming that all elements of S in Exercise 2.8 are equally likely to occur, find(a) The probability of event A:(b) The probability of event C;(c) The probability of event A DC.
A box contains 500 envelopes of which 75 contain $100 in cash, 150 contain $25, and 275 contain $10. An envelope may be purchased for $25. What is the sample space for the different amounts of money? Assign probabilities to the sample points and then find the probability that the first envelope
Suppose that in a senior college class of 500 students it is found that 210smoke, 258 drink alcoholic beverages, 216 eat between meals, 122smoke and drink alcoholic beverages, 83 eat between meals and drink alcoholic beverages, 97smoke and eat between meals, and 52 engage in all three of these bad
The probability that an American industry will locate in Shanghai, China is 0.7, the probability that it will locate in Beijing, China is 0.4, and the probability that it will locate in cither Shanghai or Beijing or both is 0.8. What is the probability that the industry will locate(a) In both
From past experiences a stockbroker believes that under present economic conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax-free bonds and mutual funds with a probability of 0.15. At
If a letter is chosen at random from the English alphabet, find the probability that the letter(a) Is a vowel exclusive of y;(b) Is listed somewhere ahead of the letter j?(c) Is listed somewhere after the letter g.
An automobile manufacturer is concerned about a possible recall of its best-selling four-door sedan. If there were a recall, there is 0.25probability that a defect is in the brake system, 0.18 in the transmission 0.17 in the fuel system, and 0.40 in some other area.(a) What is the probability that
If each coded item in a catalog begins with 3 distinct letters followed by 4 distinct nonzero digits find the probability of randomly selecting one of these coded items with the first letter a vowel and the last digit even.
A pair of fair dice is tossed. Find the probability of getting(a) A total of 8;(b) At most a total of 5.
Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 2 and less than 8?

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