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bayesian statistics an introduction

Probability & Statistics For Engineers And Scientists With R 1st Edition Michael Akritas - Solutions

The proportion of female voters who strongly support the exploration of all alternative forms of energy production is the same as the proportion of all voters who strongly support the exploration of all alternative forms of energy production. For a person selected at random from the population of
The proportion of female voters who strongly support the exploration of all alternative forms of energy production is the same as the proportion of all voters who strongly support the exploration of all alternative forms of energy production. For a person selected at random from the population of
A laundromat’s aging washing machine and clothes dryer are being replaced. The probability a new washing machine will require warranty service is 0.22. The corresponding probability for a new dryer is 0.15. What is the probability that both machines will require warranty service?
A card is drawn at random from an ordinary deck of 52 cards. Let A and B denote the events that the card is a five and the card is a spade, respectively. Are the events A and B independent?
(a) A die is rolled twice. Let A = {outcome of first roll is even} and B ={outcome of second roll is either a 1 or a 3}. Are the events A and B independent?(b) Two electronic components are selected from the production line for thorough inspection. It is known that 90% of the components have no
13. Prove the following generalization of the multiplication rule. P(EEE)=P(E1)P(E2E1)P(E3 E1 E2) P(En EnEn... En-1)..
12. Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered.Of the aircraft that are discovered, 60% have an emergency locator, whereas 10% of the aircraft not discovered have such a locator. Suppose a light aircraft disappears while in
11. Acity’s police department plans to enforce speed limits by using radar traps at four different locations. During morning rush hour, the radar traps at locations L1, L2, L3, L4 are operated with probabilities 0.4, 0.3, 0.2, and 0.3, respectively. A person speeding to work has probabilities of
10. A batch of 10 fuses contains three defective ones.A sample of size two is taken at random and without replacement.(a) Find the probability that the sample contains no defective fuses.(b) Let X be the random variable denoting the number of defective fuses in the sample. Find the probability mass
9. You ask your roommate to water a sickly plant while you are on vacation.Without water the plant will die with probability 0.8 and with water it will die with probability 0.1. With probability 0.85, your roommate will remember to water the plant.(a) What is the probability that your plant is
8. Thirty percent of credit card holders carry no monthly balance, while 70% do. Of those card holders carrying a balance, 30% have annual income $20,000 or less, 40%between $20,001 and $50,000, and 30% over $50,000.Of those card holders carrying no balance, 20%, 30%, and 50% have annual incomes in
7. Fifteen percent of all births involve Cesarean (C) section.Ninety-eight percent of all babies survive delivery, whereas when a C section is performed the baby survives with probability 0.96.(a) Make a tree diagram and mark the given information on the appropriate paths of the diagram.(Note that
6. A particular consumer product is being assembled on production lines A, B, and C, and packaged in batches of 10. Each day, the quality control team selects a production line by probability sampling with probabilities P(A) = P(B) = 0.3 and P(C) = 0.4 and inspects a randomly drawn batch from the
5. An article on vehicle choice behavior2 gives the following information about the US car and truck market.The ratio of cars to trucks is 36/64. Among the cars sold 42% are made in the US, while 58% are imports.The corresponding percentages for trucks sold are 70%and 30%.(a) Find the probability
4. Two major brands of flat screen TVs control 50% and 30% of the market, respectively. Other brands have the remaining 20% of the market. It is known that 10% of brand 1 TVs require warranty repair work, as do 20% of brand 2 and 25% of different brand TVs.(a) Find the probability that the next
3. The moisture content of batches of a chemical substance is measured on a scale from 1 to 3, while the impurity level is recorded as either low (1) or high (2).Let X and Y denote the moisture content and the impurity level, respectively, of a randomly selected batch. The probabilities for each of
2. The probability that two or more of a system’s 10 components fail between consecutive inspections is 0.005, while the probability that only one component fails is 0.1. When two or more components fail, a re-evaluation of the system is initiated during which all failed components, and those
1. The probability that a phone call will last more than t minutes is (1+t)−2. Given that a particular phone call has not ended in two minutes, what is the probability it will last more than three minutes?
Suppose that 5% of all men and 0.25% of all women are color-blind. A person is chosen at random from a community having 55% women and 45% men.(a) What is the probability that the person is color-blind?(b) If the chosen person is color-blind, what is the probability that the person is male?
In the setting of Example 2.5-11, find the probability that a passing car encountered a green first light given that it encountered a green second light.
Use the information given in Example 2.5-7 regarding the two consecutive synchronized traffic lights to complete the following.(a) Find the probability that a driver will find the second traffic light green.(b) Recalculate the probability of part (a) through a tree diagram for the experiment that
Two friends will be dealt a card each. The two cards will be drawn from a standard deck of 52 cards at random and without replacement. If neither gets an ace, the full deck is reshuffled and two cards are again drawn without replacement. The game ends when at least one of the two friends gets an
Of the customers entering a department store 30% are men and 70% are women.The probability a male shopper will spend more than $50 is 0.4, and the corresponding probability for a female shopper is 0.6. The probability that at least one of the items purchased is returned is 0.1 for male shoppers and
Pick three cards from a deck. Find the probability that the first draw is an ace, the second draw is a king, and the third draw is a queen.
Two consecutive traffic lights have been synchronized to make a run of green lights more likely. In particular, if a driver finds the first light to be red, the second light will be green with probability 0.9, and if the first light is green the second will be green with probability 0.7. If the
Fair game with an unfair coin. Suppose that when a biased coin is flipped it results in heads with probability p. A fair game with such an unfair coin can be played as follows: Flip the coin twice. If the outcome is (H, H) or (T, T) ignore the outcome and flip the coin two more times. Repeat until
A supermarket has regular (manned) checkout lines and self checkout lines. Let X take the value 0, 1, or 2 depending on whether there are no customers, between 1 and 10 customers, or more than 10 customers, respectively, in the self checkout lines in any given five-minute period. Let Y be the
The probability that the life of a product does not exceed t time units is 1 − exp(−0.1t). Given that a product has lasted 10 time units, what is the probability it will fail in the next 5 time units?
Example 2.4-3 gave the percentages of families in a certain community that own a dog, a cat, or both as 60%, 70%, and 50%, respectively. If a randomly selected family owns a dog, what is the probability it also owns a cat?
Two dice are rolled and their sum is observed to be 7. Given this information, what is the conditional probability that one of the two die rolls was a 3?
A card drawn at randomfroma deck of 52 cards is observed to be a face card.Given this partial information, what is the probability the card is a king?
12. Let’s make a deal. In the game Let’s Make a Deal, the host asks a participant to choose one of three doors. Behind one of the doors is a big prize (e.g., a car), while behind the other two doors are minor prizes (e.g., a blender). After the participant selects a door, the host opens one of
11. Efron’s dice. Using arbitrary numbers on the sides of dice can have surprising consequences for the game described in the Exercise 10. Efron’s dice are sets of dice with the property that for each die there is another that beats it with larger probability when the game of Exercise 10 is
10. Consider the game where two dice, die A and die B, are rolled. We say that die A wins, and write A > B, if the outcome of rolling A is larger than that of rolling B. If both rolls result in the same number it is a tie.(a) Find the probability of a tie.(b) Find the probability that die A wins.
9. A type of communications systemworks if at least half of its components work. Suppose it is possible to add a fifth component to such a system having four components.Show that the resulting five-component system is not necessarily more reliable. (Hint. In the notation of Example 2.4-2, it
8. The monthly volume of book sales from the online site of a bookstore is categorized as below expectations (0), in line with expectations (1), or above expectations (2). The monthly volume of book sales from the brick and mortar counterpart of the bookstore is categorized similarly. The following
7. The electronics fabrication plant in Exercise 6 has a third machine, machine C, which is used in periods of peak demand and is also capable of producing a batch of 50 electrical components per hour. Let E1, E2 be as in Exercise 6, and E3 be the corresponding event for machine C. The
6. Each of the machines A and B in an electronics fabrication plant produces a single batch of 50 electrical components per hour. Let E1 denote the event that, in any given hour, machine A produces a batch with no defective components, and E2 denote the corresponding event for machine B. The
5. A telecommunications company classifies transmission calls by their duration as brief ((> 3 minutes) and by their type as voice (V), data (D), or fax (F). From an extensive data base, the company has come up with the following probabilities for the category of a random (e.g. the next)
4. The PMF of the sum of two die rolls, found in Example 2.3-13, is(a) For each of the following events specify the outcomes that belong to them, and use relation (2.4.1) to find their probabilities.(i) E1 = {the sum of the two die rolls is at least 5}.(ii) E2 = {the sum of the two die rolls is no
3. A new generation of hybrid vehicles achieves highway gas mileage in the range of 50 to 53 MPG. The gas mileage of three such cars, during a pre-specified 100-mile drive, will be rounded to the nearest integer, resulting in the sample space S = {(x1, x2, x3) : xi = 50, 51, 52, 53}, i = 1, 2,
2. The events A1,A2, . . . ,Am are said to form a partition of the sample space of an experiment if they are disjoint and their union equals the sample space.(a) If all m events of such a partition are equally likely, what is their common probability?(b) If m = 8 find the probability of A1 ∪ A2
1. A person is selected at randomfromthe population of Verizon wireless subscribers. Let A be the event that the chosen subscriber has friends or family added to his/her plan, and B denote the event that the subscriber has unlimited text messaging. Extensive records suggest that P(A) = 0.37, P(B) =
In a certain community, 60% of the families own a dog, 70% own a cat, and 50% own both a dog and a cat. If a family is selected at random, calculate the probabilities of the following events.(a) The family owns a dog but not a cat.(b) The family owns a cat but not a dog.(c) The family owns at least
The reliability of a system is defined as the probability that a system will function correctly under stated conditions. A system’s reliability depends on the reliability of its components as well as the way the components are arranged. A type of communications system works if at least half of
The PMF of the experiment that records the number of heads in four flips of a coin, which can be obtained with the R commands attach(expand.grid(X1=0:1, X2=0:1, X3=0:1, X4=0:1)); table(X1+X2+X3+X4)/length(X1), isThus, if the random variable X denotes the number of heads in four flips of a coin then
19. The multinomial theorem states that n (a+...+ar)"= (m....).... n1,n2.. nr where denotes the sum over all nonnegative n++n=n integers n,2,...,nr, which sum to n. Using the multi- nomial theorem, expand (a + 2a2+ a3).
18. The binomial theorem states that n n akb bn-k (a+b)"= =(") k=0
17. After testing, asphalt shingles are classified as high, medium, or low grade, and different grade shingles are sold under different warranties.(a) In how many ways can the next 15 shingles be classified into high, medium, or low grade? (Hint. Think of a task consisting of 15 stages, with each
16. The information technology company mentioned in Exercise 7 has 10 EE majors working as interns.Two interns will be assigned to work on each of the four programing projects, and the remaining two interns will be assigned to another project. How many possible assignments of the 10 interns are
15. A simple random sample of size five is selected from 52 cards. Find the probability of each of the following events.(a) The five-card hand contains all four 2’s.(b) The five-card hand contains two aces and two 7’s.(c) The five-card hand contains three kings and the other two cards are of
14. A forest contains 30 moose ofwhich six are captured, tagged, and released. A certain time later, five of the 30 moose are captured.(a) How many samples of size five are possible?(b) How many samples of size five, which include two of the six tagged moose, are possible?(c) If the five captured
13. Five of the 15 school buses of a particular school district will be selected for thorough inspection. Suppose four of the buses have developed a slight defect since their last inspection (the steering wheel shakes when braking).(a) How many possible selections are there?(b) How many selections
12. A communication system consists of 13 antennas arranged in a line. The system functions as long as no two nonfunctioning antennas are next to each other. Suppose five antennas stop functioning.(a) How many different arrangements of the five nonfunctioning antennas result in the system being
11. A path going fromthe lower left corner of the grid in Figure 2-7 to the upper right corner can be represented as a binary sequence having four 0’s and four 1’s,with each 0 representing a move to the right and each 1 representing a move upwards.(a) How many paths going from the lower left
10. Answer the following questions.(a) A team of 5 starters will be selected from 10 basketball players. How many selections are there?(b) Ten basketball players will be divided into two teams for a practice game. How many divisions of the 10 players into two teams of 5 are there?(c) If each of 12
9. Twelve individuals want to form a committee of four.(a) How many committees are possible?(b) The 12 individuals consist of 5 biologists, 4 chemists, and 3 physicists. How many committees consisting of 2 biologists, 1 chemist, and 1 physicist are possible?(c) In the setting of part (b), if all
8. In many countries the license plates consist of a string of seven characters such that the first three are letters and the last four are numbers. If each such string of seven characters is equally likely, what is the probability that the string of three letters begins with a W and the string of
7. An information technology company will assign four electrical engineers to four different JAVA programming projects (one to each project). How many different assignments are there?
6. A test consists of five true-false questions.(a) In how many ways can it be completed? (Hint. The task of answering five true-false questions consists of five stages.)(b) A correctly answered question receives 1 point, while an incorrectly answered question gets 0. Give the sample space for the
5. The R command S=expand.grid(X1=1:6, X2=1:6) generates the sample space of two die rolls. The additional R command attach(S); Y=(X1==6)+(X2==6) generates the number of times a six occurs for each of the 36 outcomes in the sample space. Finally, the additional R commands pr=table(Y)/36; pr
4. The following two questions pertain to the spray tank irrigator of Exercise 3.(a) The water pH level is measured over the next two irrigations.(i) Give the sample space of this experiment, and its size, using R commands t=seq(6.8, 7.2, 0.1);G=expand.grid(X1=t,X2=t); G; length(G$X1).(ii) Give the
3. Soil or water pH is measured on a scale of 0–14. A pH reading below 7 is considered acidic, while a pH reading above 7 is basic. The pH level of water provided by a type of spray tank irrigator is equally likely to be either 6.8, 6.9, 7.0, 7.1 or 7.2. Let E1 denote the event that at the next
2. Refer to Exercise 1.(a) Select two wafers, at random and with replacement, from the batch of 10 wafers given in the exercise.(i) Give the sample space for the experiment that records the doping type of the two wafers and the probability for each outcome.(ii) Give the sample space of the
1. In electronics, a wafer is a thin slice of semiconductor material used in the fabrication of integrated circuits and other micro-devices. They are formed of highly pure crystalline material, which is doped (i.e., impurity atoms are added) to modulate its electrical properties. The doping is
Simulating an experiment with R. Use the PMF obtained in Example 2.3-13 to simulate 10 repetitions of the experiment that records the sum of two die rolls.
A simple random sample of size n = 3 is drawn from a batch of 10 product items.If three of the 10 items are defective, find the PMF of the random variable X ={number of defective items in the sample}.
The clock rate of a CPU (central processing unit) chip refers to the frequency, measured in megahertz (MHz), at which it functions reliably. CPU manufacturers typically categorize (bin) CPUs according to their clock rate and charge more for CPUs that operate at higher clock rates. A chip
In the game of poker each player receives five cards dealt from a deck of 52 cards.Full house refers to a five-card hand consisting of three of a kind and two of a kind.An example of a full house is a hand consisting of three 10’s and two 5’s. Find the probability that a randomly dealt
(a) How many binary sequences of length 10 with exactly four 1’s that are not consecutive can be formed?(b) If a binary sequence of length 10 having four 1’s is selected at random, what is the probability that the four 1’s are nonconsecutive?
(a) How many binary sequences (i.e., sequences of 0’s and 1’s) of length 10 with exactly four 1’s can be formed?(b) If a binary sequence of length 10 is selected at random, what is the probability that it has four 1’s?
Two cards will be selected from a deck of 52 cards.(a) How many outcomes are there if the first card will be given to player 1 and the second card will be given to player 2?(b) How many outcomes are there if both cards will be given to player 1?
(a) The lineup or batting order is a list of the nine baseball players for a team in the order they will bat during the game. How many lineups are possible?(b) The Department of Tranpostation (DOT) plans to assign six civil engineers to oversee six interstate safety design projects. How many
In selecting a first and second place winner from the group of four finalists consisting of Niki, George, Sophia, and Martha, the outcomes (George, Sophia) and(Sophia, George) are two of the 12 distinct outcomes mentioned in Example 2.3-3(c). However, if both winners will receive the same prize, so
For each of the following tasks specify the stages, the number of outcomes of each stage and the number of outcomes of the task.(a) Select a plumber, an electrician, and a remodeler from three plumbers, two electricians, and four remodelers available in the yellow pages.(b) Form a binary sequence
For each of the following tasks specify the stages, the number of outcomes of each stage, and the number of outcomes of the task.(a) Select a plumber and an electrician from three plumbers and two electricians available in the yellow pages.(b) Select two items from an assembly line, and classify
Roll two dice separately (or one die twice). Find the probability of the event that the sum of the two sides is seven.
The efficiency of laser diodes (measured at 25oC in mW per mA) varies from 2 to 4. In a shipment of 100, the numbers having efficiency 2, 2.5, 3, 3.5, and 4 are 10, 15, 50, 15, and 10, respectively. One laser diode is randomly selected. Find the probabilities of the events E1 = {the selected laser
10. A random sample of 100 polycarbonate plastic disks are categorized according to their hardness and shock absorption. The results are shown in the table below.Adisk is selected at random.Define the events E1 = {the disk has low hardness}, E2 = {the disk has low shock absorption}, E3 = {the disk
9. In measuring the diameter of a cylinder to the nearest millimeter the sample space (in cm) is S ={5.3, 5.4, 5.5, 5.6, 5.7}. Five cylinders are randomly selected and their diameters are measured to the nearest millimeter. (a) Describe the sample space of this experiment.What is the size of this
8. Prove that the pairs of event given in Exercises 4, 6, and 7 are equal by means of a sequence of formal (logical)arguments showing that, in each case, each event is a subset of the other.
7. Sketch two Venn diagrams like the one in Figure 2-4.On the first shade the event (A∩B)∪C, and on the second shade the event (A ∪ C) ∩ (B ∪ C). Are they the same? A B C Figure 2-4 Generic Venn diagram with three events.
6. Prove the second of De Morgan’s Laws by sketching two Venn diagrams like the one in Figure 2-3. On the first shade the event (A ∩ B)c, on the second shade the event Ac ∪ Bc, and then confirm that they are the same.
5. In testing the lifetime of components, the sample space is the set S = {x:x > 0} of positive real numbers. Let A be the event that the next component tested will last less than 75 time units and B the event that it will last more than 53 time units. In mathematical notation, A = {x:x 53}.
4. Sketch two Venn diagrams like the one in Figure 2-3.On the first shade the set (A − B) ∪ (B − A), and on the second shade the event (A ∪ B) − (A ∩ B). Are they the same? A B Figure 2-3 Generic Venn diagram with two events.
3. An engineering firm is considering the possibility of establishing a branch office in Toronto and one in Mexico City. Let T be the event that the firm will establish a branch office in Toronto andM be the event that the firm will establish a branch office in Mexico City.(a) Express each of the
2. In a certain community, 40% of the households subscribe to a local newspaper, 30% subscribe to a newspaper of national circulation, and 60% subscribe to at least one of the two types of newspapers. Let E1 denote the event that a randomly chosen household subscribes to a local newspaper, and let
1. Give the sample space for each of the following experiments.(a) A die is rolled twice and the outcomes are recorded.(b) A die is rolled twice and the sum of the outcomes is recorded.(c) From a shipment of 500 iPods, 6 of which have a click wheel problem, a simple random sample of 30 ipods is
In telecommunications, a handoff or handover is when a cellphone call in progress is redirected from its current cell (called source) to a new cell (called target). For example, this may happen when the phone is moving away from the area covered by the source cell and entering the area covered by
In measuring the diameter of a cylinder the sample space (in cm) is S = {x : 5.3 ≤ x ≤ 5.7}. Let E1 = {x : x > 5.4} and E2 = {x : x < 5.6}. Describe the events E1 ∪ E2, E1 ∩ E2, and E1 − E2.
Resistors manufactured by a machine producing 1k& resistors are unacceptable if they are not within 50& of the nominal value. Four such resistors are tested.(a) Describe the sample space of this experiment.(b) Let Ei denote the event that the ith resistor tests acceptable. Are the Eis mutually
Three undergraduate students from a particular university are selected and their opinions about a proposal to expand the use of solar energy are recorded on a scale from 1 to 10.(a) Describe the sample space of this experiment. What is the size of this sample space?(b) Describe the sample space if
An undergraduate student from a particular university is selected and his/her opinion about a proposal to expand the use of solar energy is recorded on a scale from 1 to 10.(a) Give the sample space of this experiment.(b) How does the sample space differ from the statistical population?
(a) Give the sample space of the experiment that selects two fuses and classifies each as non-defective or defective.(b) Give the sample space of the experiment that selects two fuses and records how many are defective.(c) Give the sample space of the experiment that records the number of fuses
In 20 flips of a coin, 18 heads result. Should the fairness of the coin be dismissed?
(a) If 75% of citizens of voting age are in favor of introducing incentives for the use of solar energy, what are the chances that in a sample of 1000, at least 650 will be in favor of such incentives?(b) If 5% of electrical components have a certain defect, what are the chances that a batch of 500
(a) What are the chances of one heads in two flips of a fair coin? This can be thought of as taking a sample of size two with replacement from the population consisting of {Heads, Tails} and asking for the chances of the sample containing the item heads only once.(b) What are the chances of 4
19. The data file AdLocNews.txt contains the number of inquiries regarding ads placed in a local newspaper.The ads are categorized according to the day of the week and in which section of the newspaper they appeared. Use R to:(a) Compute the cell means and the main and interaction effects for the
18. The file SpruceMothTrap.txt contains data on the number of moths caught in moth traps using different lures and placed at different locations on spruce trees.22 Use R to:(a) Compute the cell means and the main and interaction effects for the factors location and lure.(b) Construct an
17. A soil scientist is considering the effect of soil pH level on the breakdown of a pesticide residue. Two pH levels are considered in the study. Because pesticide residue breakdown is affected by soil temperature, four different temperatures are included in the study.(a) Draw the interaction

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