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mathematics
statistics

John E Freunds Mathematical Statistics With Applications 8th Edition Irwin Miller, Marylees Miller - Solutions

At an electronics plant, it is known from past experience that the probability is 0.84 that a new worker who has attended the company’s training program will meet the production quota, And that the corresponding probability is 0.49 for a new worker who has not attended the company’s training
It is known from experience that in a certain industry 60 percent of all labor– management disputes are over wages, 15 percent are over working conditions, And 25 percent are over fringe issues. Also, 45 percent of the disputes over wages are resolved without strikes, 70 percent of the disputes
In a T- maze, a rat is given food if it turns left And an electric shock if it turns right. On the first trial there is a 50– 50 chance that a rat will turn either way; then, if it receives food on the first trial, the probability is 0.68 that it will turn left on the next trial, And if it
The probability that a one- car accident is due to faulty brakes is 0.04, the probability that a one- car accident is correctly attributed to faulty brakes is 0.82, And the probability that a one- car accident is incorrectly attributed to faulty brakes is 0.03. What is the probability that (a) A
With reference to Example 2.25, suppose that we discover later that the job was completed on time. What is the probability that there had been a strike? In exercise The completion of a construction job may be delayed because of a strike. The probabilities are 0.60 that there will be a strike, 0.85
In a certain community, 8 percent of all adults over 50 have diabetes. If a health service in this community correctly diagnoses 95 percent of all persons with diabetes as having the disease And incorrectly diagnoses 2 percent of all persons without diabetes as having the disease, find the
An explosion at a construction site could have occurred as the result of static electricity, malfunctioning of equipment, carelessness, or sabotage. Interviews with construction engineers analyzing the risks involved led to the estimates that such an explosion would occur with probability 0.25 as a
Give an alternative proof of Theorem 2.7 by making use of the relationships A ∪ B = A ∪ (A' ∩ B') And B = (A ∩ B) .(A' ∩ B) .
An art dealer receives a shipment of five old paintings from abroad, And, on the basis of past experience, she feels that the probabilities are, respectively, 0.76, 0.09, 0.02, 0.01, 0.02, And 0.10 that 0, 1, 2, 3, 4, or all 5 of them are forgeries. Since the cost of authentication is fairly high,
To get answers to sensitive questions, we some-times use a method called the randomized response technique. Suppose, for instance, that we want to determine what percentage of the students at a large university smoke marijuana. We construct 20 flash cards, write “ I smoke marijuana at least once
Use the Venn diagram of Figure 2.7 And the method by which we proved Theorem 2.7 to prove Theorem 2.8.Figure 2.7
A series system consists of two components having the reliabilities 0.98 And 0.99, respectively, connected to a parallel subsystem containing five components having the reliabilities 0.75, 0.60, 0.65, 0.70, And 0.60, respectively. Find the system reliability.
Duplicate the method of proof used in Exercise 2.12 to show that P(A ∪ B ∪ C ∪ D) = P(A) + P(B) + P(C) + P(D) - P(A ∩ B) - P(A ∩ C) - P(A ∩ D) - P(B ∩ C) - P(B ∩ D) - P(C ∩ D) + P(A ∩ B ∩ C) + P(A ∩ B ∩ D) + P(A ∩ C ∩ D) + P(B ∩ C ∩ D) - P(A ∩ B ∩ C ∩ D)
Prove by induction thatFor Any finite sequence of events E1, E2, ∪ . ., And En.
The odds that an event will occur are given by the ratio of the probability that the event will occur to the probability that it will not occur, provided neither probability is zero. Odds are usually quoted in terms of positive integers having no common factor. Show that if the odds are A to B that
Subjective probabilities may be determined by exposing persons to risk- taking situations And finding the odds at which they would consider it fair to bet on the outcome. The odds are then converted into probabilities by means of the formula of Exercise 2.15. For instance, if a person feels that 3
Show that the postulates of probability are satisfied by conditional probabilities. In other words, show that if P(B) ≠ 0, then (a) P(A| B) ≥ 0; (b) P(B| B) = 1; (c) P(A1 ∪ A2 .. ∪ . | B) = P(A1| B) + P(A2| B) + · · · for Any sequence of mutually exclusive events A1, A2, ∪ . ..
Show by means of numerical examples that P(B| A) + P(B| A') (a) May be equal to 1; (b) Need not be equal to 1.
Duplicating the method of proof of Theorem 2.10, show that P(A ∩ B ∩ C ∪ D) = P(A) · P(B| A) · P(C| A ∩ B) · P(D| A ∩ B ∩ C) provided that P(A ∩ B ∩ C) ≠ 0.
Given three events A, B, And C such that P(A ∩ B ∩ C) Z0 And P(C| A ∩ B) = P(C| B) , show that P(A| B ∩ C) = P(A| B) .
Show that if P(B| A) = P(B) And P(B) ≠ 0, then P(A| B) = P(A) .
Show that if events A And B are independent, then (a) Events A' And B are independent; (b) Events A' And B' are independent.
Show that if events A And B are dependent, then events A And B' are dependent.
Refer to Figure 2.10 to show that P(A ˆ© B ˆ© C) = P(A) · P(B) · P(C) does not necessarily imply that A, B, And C are all pairwise independent.Figure 2.10
If events A, B, And C are independent, show that (a) A And B ∩ C are independent; (b) A And B ∪ C are independent.
If A1, A2,…, An are independent events, prove that P(A1 ∪ A2,···, An) = 1-{1- P(A1) } · {1- P(A2) } … {1- P(A ∩) }
Show that 2k - k- 1 conditions must be satisfied for k events to be independent.
For Any event A, show that A And Ø are independent.
Prove Theorem 2.12 by making use of the following generalization of the distributive law given in part (b) of Exercise 2.1: A ∩(B1. B2. · · · ∪ Bk) =(A ∩ B1) .(A ∩ B2) ∪ · · · ∪ (A ∩ Bk)
Suppose that a die has n sides numbered i = 1, 2,···, n. Assume that the probability of it coming up on the side numbered i is the same for each value of i. The die is rolled n times (assume independence) And a “ match” is defined to be the occurrence of side i on the ith roll. Prove that
Show that P(A ∪ B) ≥ 1- P(A') - P(B') for Any two events A And B defined in the sample space S.
If S = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 3, 5, 7}, B = {6, 7, 8, 9}, C = {2, 4, 8}, And D = {1, 5, 9}, list the elements of the subsets of S corresponding to the following events: (a) A ∩ B; (b) (A ∩ B) ∩ C; (c) B ∪ C; (d) (B ∪ C) ∩ D; (e) A ∩ C; (f) (A ∩ C) ∩ D.
An electronics firm plans to build a research lab-oratory in Southern California, And its management has to decide between sites in Los Angeles, San Diego, Long Beach, Pasadena, Santa Barbara, Anaheim, Santa Monica, And Westwood. If A represents the event that they will choose a site in San Diego
Among the eight cars that a dealer has in his show-room, Car 1 is new And has air- conditioning, power steering, And bucket seats; Car 2 is one year old And has air-conditioning, but neither power steering nor bucket seats; Car 3 is two years old And has air- conditioning And power steering, but no
With reference to Exercise 2.37, state in words what kind of car the customer will choose, if his choice is given by (a) The complement of the set of part (a); (b) The union of the sets of parts (b) And (c); (c) The intersection of the sets of parts (c) And (d); (d) The intersection of parts (b)
If Ms. Brown buys one of the houses advertised for sale in a Seattle newspaper (on a given Sunday) , T is the event that the house has three or more baths, U is the event that it has a fireplace, V is the event that it costs more than $ 200,000, And W is the event that it is new, describe (in
A coin is tossed once. Then, if it comes up heads, a die is thrown once; if the coin comes up tails, it is tossed twice more. Using the notation in which (H, 2) , for example, denotes the event that the coin comes up heads And then the die comes up 2, And (T, T, T) denotes the event that the coin
An electronic game contains three components arranged in the series– parallel circuit shown in Figure 2.12. At Any given time, each component may or may not be operative, And the game will operate only if there is a continuous circuit from P to Q. Let A be the event that the game will operate;
An experiment consists of rolling a die until a 3 appears. Describe the sample space And determine (a) How many elements of the sample space correspond to the event that the 3 appears on the kth roll of the die; (b) How many elements of the sample space correspond to the event that the 3 appears
If S = {x| 0< x< 10}, M = {x| 3< x ≤ 8}, And ∩ = {x| 5< x< 10}, find (a) M. N; (b) M ∩ N; (c) M ∩ N' ; (d) M ∪ N.
In Figure 2.13, L is the event that a driver has liability insurance And C is the event that she has collision insurance. Express in words what events are represented by regions 1, 2, 3, And 4.
With reference to Exercise 2.46 And Figure 2.13, what events are represented by (a) Regions 1 And 2 together; (b) Regions 2 And 4 together;(c) Regions 1, 2, And 3 together; (d) Regions 2, 3, And 4 together?In exerciseIn Figure 2.13, L is the event that a driver has liability
In Figure 2.14, E, T, And N are the events that a car brought to a garage needs an engine overhaul, trans-mission repairs, or new tires. Express in words the events represented by (a) Region 1; (b) Region 3; (c) Region 7; (d) Regions 1 And 4 together; (e) Regions 2 And 5
With reference to Exercise 2.48 And Figure 2.14, list the region or combinations of regions representing the events that a car brought to the garage needs (a) Transmission repairs, but neither An engine overhaul nor new tires; (b) AN engine overhaul And transmission repairs; (c)
A market research organization claims that, among 500 shoppers interviewed, 308 regularly buy Product X, 266 regularly buy Product Y, 103 regularly buy both, And 59 buy neither on a regular basis. Using a Venn diagram And filling in the number of shoppers associated with the various regions, check
In a group of 200 college students, 138 are enrolled in a course in psychology, 115 are enrolled in a course in sociology, And 91 are enrolled in both. How many of these students are not enrolled in either course?
Among 120 visitors to Disneyland, 74 stayed for at least 3 hours, 86 spent at least $ 20, 64 went on the Matterhorn ride, 60 stayed for at least 3 hours And spent at least $ 20, 52 stayed for at least 3 hours And went on the Matterhorn ride, 54 spent at least $ 20 And went on the Matterhorn ride,
An experiment has five possible outcomes, A, B, C, D, And E, that are mutually exclusive. Check whether the following assignments of probabilities are permissible And explain your Answers: (a) P(A) = 0.20, P(B) = 0.20, P(C) = 0.20, P(D) = 0.20, And P(E) = 0.20; (b) P(A) = 0.21, P(B) = 0.26, P(C) =
If A And B are mutually exclusive, P(A) = 0.37, And P(B) = 0.44, find (a) P(A); (b) P(B);(c) P(A ∪ B);(d) P(A ∩ B);(e) P(A ∩ B);(f) P(A ∩ B).
Explain why there must be a mistake in each of the following statements: (a) The probability that Jean will pass the bar examination is 0.66 And the probability that she will not pass is - 0.34. (b) The probability that the home team will win An upcoming football game is 0.77, the probability that
The probabilities that the serviceability of a new X-ray machine will be rated very difficult, difficult, aver-age, easy, or very easy are, respectively, 0.12, 0.17, 0.34, 0.29, And 0.08. Find the probabilities that the serviceability of the machine will be rated (a) Difficult or very difficult;
Suppose that each of the 30 points of the sample space of Exercise 2.40 is assigned the probability 1/30. Find the probabilities that at a given moment (a) At least one of the station wagons is empty; (b) Each of the two station wagons carries the same number of passengers; (c) The larger
A hat contains 20 white slips of paper numbered from 1 through 20, 10 red slips of paper numbered from 1 through 10, 40 yellow slips of paper numbered from 1 through 40, and 10 blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has
A police department needs new tires for its patrol cars And the probabilities are 0.15, 0.24, 0.03, 0.28, 0.22, And 0.08, respectively, that it will buy Uniroyal tires, Goodyear tires, Michelin tires, General tires, Goodrich tires, or Armstrong tires. Find the probabilities that it will buy (a)
Referring to Figure 2.6, verify that P(A ∩ B') = P(A) - P(A ∩ B) Figure 2.6
Four candidates are seeking a vacancy on a school board. If A is twice as likely to be elected as B, And B And C are given about the same chance of being elected, while C is twice as likely to be elected as D, what are the probabilities that (a) C will win; (b) A will not win?
In a poker game, five cards are dealt at random from an ordinary deck of 52 playing cards. Find the probabilities of getting (a) Two pairs (Any two distinct face values occurring exactly twice); (b) Four of a kind (four cards of equal face value) .
In a game of Yahtzee, five balanced dice are rolled simultaneously. Find the probabilities of getting (a) Two pairs; (b) Three of a kind; (c) A full house (three of a kind And a pair); (d) Four of a kind.
Explain on the basis of the various rules of Exercises 2.5 through 2.9 why there is a mistake in each of the following statements: (a) The probability that it will rain is 0.67, And the probability that it will rain or snow is 0.55.(b) The probability that a student will get a passing grade in
Among the 78 doctors on the staff of a hospital, 64 carry malpractice insurance, 36 are surgeons, And 34 of the surgeons carry malpractice insurance. If one of these doctors is chosen by lot to represent the hospital staff at A ∩ A ∪ M. A ∪ convention (that is, each doctor has a probability
A right triangle has the legs 3 And 4 units, respectively. Find the probability that a line segment, drawn at random parallel to the hypotenuse And contained entirely in the triangle, will divide the triangle so that the area between the line And the vertex opposite the hypotenuse will equal at
Given P(A) = 0.59, P(B) = 0.30, And P(A ∩ B) = 0.21, find (a) P(A ∪ B) ; (b) P(A ∩ B') ; (c) P(A' ∪ B') ; (d) P(A' ∩ B') .
Referring to Figure 2.6 And letting P(A' ∩ B') = d, verify thatP(A' ∩ B') = 1 - P(A) - P(B) + P(A ∩ B)Figure 2.6
At Roanoke College it is known that 13 of the students live off campus. It is also known that 59 of the students are from within the state of Virginia And that 34 of the students are from out of state or live on campus. What is the probability that a student selected at random from Roanoke College
A biology professor has two graduate assistants helping her with her research. The probability that the older of the two assistants will be absent on Any given day is 0.08, the probability that the younger of the two will be absent on Any given day is 0.05, And the probability that they will both
Suppose that if a person travels to Europe for the first time, the probability that he will see London is 0.70, the probability that he will see Paris is 0.64, the probability that he will see Rome is 0.58, the probability that he will see Amsterdam is 0.58, the probability that he will see London
Use the formula of Exercise 2.15 to convert each of the following odds to probabilities: (a) If three eggs are randomly chosen from a carton of 12 eggs of which 3 are cracked, the odds are 34 to 21 that at least one of them will be cracked. (b) If a person has eight $ 1 bills, five $ 5 bills, And
Use the definition of “odds” given in Exercise 2.15 to convert each of the following probabilities to odds: (a) The probability that the last digit of a car’s license plate is a 2, 3, 4, 5, 6, or 7 is 6/10. (b) The probability of getting at least two heads in four flips of a balanced coin is
There are 90 applicants for a job with the news department of a television station. Some of them are college graduates And some are not; some of them have at least three years’ experience And some have not, with the exact breakdown beingIf the order in which the applicants are interviewed by the
With reference to Exercise 2.68, what is the probability that a husband will vote in the election given that his wife is going to vote? In exercise For married couples living in a certain suburb, the probability that the husband will vote in a school board election is 0.21, the probability that the
With reference to Exercise 2.70, what is the probability that one of the students will be living on campus given that he or she is from out of state? In exercise At Roanoke College it is known that 13 of the students live off campus. It is also known that 59 of the students are from within the
A bin contains 100 balls, of which 25 are red, 40 are white, And 35 are black. If two balls are selected from the bin without replacement, what is the probability that one will be red And one will be white?
If subjective probabilities are determined by the method suggested in Exercise 2.16, the third postulate of probability may not be satisfied. However, proponents of the subjective probability concept usually impose this postulate as a consistency criterion; in other words, they regard subjective
If we let x = the number of spots facing up when a pair of dice is cast, then we can use the sample space S2 of Example 2.2 to describe the outcomes of the experiment. (a) Find the probability of each outcome in S2. (b) Verify that the sum of these probabilities is 1.
Using a computer program that can generate random integers on the interval (0, 9) with equal probabilities, generate 1,000 such integers And use the frequency interpretation to estimate the probability that such a randomly chosen integer will have a value less than 1.
Using the method of Exercise 2.85, generate a second set of 1,000 random integers on (0, 9). Estimate the probability that A: an integer selected at random from the first set will be less than 1 or B: An integer selected at random from the second set will be less than 1 (a) Using the frequency
With reference to Exercise 2.72, find the probabilities that a person who visits Disneyland will (a) Ride the Monorail given that he will go on the Jungle Cruise;(b) Go on the Matterhorn ride given that he will go on the Jungle Cruise And ride the Monorail; (c) Not go on the Jungle Cruise
Crates of eggs are inspected for blood clots by randomly removing three eggs in succession And examining their contents. If all three eggs are good, the crate is shipped; otherwise it is rejected. What is the probability that a crate will be shipped if it contains 120 eggs, of which 10 have blood
Use the formula of Theorem 2.7 to show that In Theorem 2.7 P(A ∪ B) = P(A) + P(B) – P(An B) (a) P(A ∩ B) ≤ P(A) + P(B); (b) P(A ∩ B) ≥ P(A) + P(B) - 1.
Suppose that in Vancouver, B ∪ C., the probability that a rainy fall day is followed by a rainy day is 0.80 And the probability that a sunny fall day is followed by a rainy day is 0.60. Find the probabilities that a rainy fall day is followed by (a) A rainy day, a sunny day, And An other rainy
A sharpshooter hits a target with probability 0.75. Assuming independence, find the probabilities of getting (a) A hit followed by two misses; (b) Two hits And a miss in Any order.
A balanced die is tossed twice. If A is the event that an even number comes up on the first toss, B is the event that an even number comes up on the second toss, And C is the event that both tosses result in the same number, are the events A, B, And C (a) Pairwise independent; (b) Independent?
A shipment of 1,000 parts contains 1 percent defective parts. Find the probability that (a) The first four parts chosen arbitrarily for inspection are nondefective; (b) The first defective part found will be on the fourth inspection.
A coin is loaded so that the probabilities of heads And tails are 0.52 And 0.48, respectively. If the coin is tossed three times, what are the probabilities of getting (a) All heads; (b) Two tails And a head in that order?
For each of the following, determine whether the given values can serve as the values of a probability distribution of a random variable with the range x = 1, 2, 3, and 4: (a) f (1) = 0.25, f (2) = 0.75, f (3) = 0.25, and f (4) = - 0.25; (b) f (1) = 0.15, f (2) = 0.27, f (3) = 0.29, and f (4) =
Find the distribution function of the random variable of part (a) of Exercise 3.7 and plot its graph.
A sharpshooter is aiming at a circular target with radius 1. If we draw a rectangular system of coordinates with its origin at the center of the target, the coordinates of the point of impact, (X, Y), are random variables having the joint probability density Find (a) P[(X,Y) ϵ A], where
Suppose that P, the price of a certain commodity (in dollars), and S, its total sales (in 10,000 units), are random variables whose joint probability distribution can be approximated closely with the joint probability densityFind the probabilities that (a) The price will be less than 30 cents
A certain college gives aptitude tests in the sciences and the humanities to all entering freshmen. If X and Y are, respectively, the proportions of correct answers that a student gets on the tests in the two subjects, the joint probability distribution of these random variables can be approximated
With reference to Exercise 3.97, In exercise Two textbooks are selected at random from a shelf that contains three statistics texts, two mathematics texts, and three physics texts. If X is the number of statistics texts and Y the number of mathematics texts actually chosen, construct a table
If X is the proportion of persons who will respond to one kind of mail- order solicitation, Y is the proportion of persons who will respond to another kind of mail- order solicitation, and the joint probability density of X and Y is given byFind the probabilities that(a) At least 30 percent will
If two cards are randomly drawn (without replacement) from an ordinary deck of 52 playing cards, Z is the number of aces obtained in the first draw, and W is the total number of aces obtained in both draws, find (a) The joint probability distribution of Z and W; (b) The marginal distribution of
With reference to Exercise 3.101, In exerciseSuppose that P, the price of a certain commodity (in dollars), and S, its total sales (in 10,000 units), are random variables whose joint probability distribution can be approximated closely with the joint probability densityFind (a) The
If X is the amount of money (in dollars) that a salesperson spends on gasoline during a day and Y is the corresponding amount of money (in dollars) for which he or she is reimbursed, the joint density of these two random variables is given byFind (a) The marginal density of X; (b) The
Show that the two random variables of Exercise 3.102 are not independent.In exerciseA certain college gives aptitude tests in the sciences and the humanities to all entering freshmen. If X and Y are, respectively, the proportions of correct answers that a student gets on the tests in the two
The useful life (in hours) of a certain kind of integrated circuit is a random variable having the probability densityIf three of these circuits operate independently, Find (a) The joint probability density of X1, X2, and X3, representing the lengths of their useful lives; (b) The value
If X has the distribution functionFind(a) P(2< X ≤ 6); (b) P(X = 4); (c) The probability distribution of X.
The following are the percentages of tin in measurements made on 24 solder joints:(a) Construct a stem- and- leaf diagram using 5 and 6 as the stem labels. (b) Construct a double- stem display.(c) Which is more informative?
Suppose the first row of 12 observations in Exercise 3.110 came from solder connections made at station 105 and the second row came from station 107. Use a pair of stem- and- leaf diagrams to determine whether you should suspect a difference in the soldering process at the two stations.
Two different lathes turn shafts to be used in electric motors. Measurements made of their diameters (in cm) areConstruct two stem-and- leaf diagrams to see if you should suspect that the two lathes are turning out shafts of different diameters.

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