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bayesian statistics an introduction

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

1. In an accelerated life testing experiment, different batches of n equipment are operated under different stress conditions. Because the stress level is randomly set for each batch, the probability, P, with which an equipment will last more than T time units is a random variable.In this problem
In the context of Example 4.6-5, set N1 for the number of these components that last less than 50 hours, N2 for the number that last between 50 and 90 hours, and N3 for the number that last more than 90 hours.(a) Find the probability that exactly one of the eight electronic components will last
The probabilities that a certain electronic component will last less than 50 hours in continuous use, between 50 and 90 hours, or more than 90 hours, are p1 = 0.2, p2 = 0.5, and p3 = 0.3, respectively. The time to failure of eight such electronic components will be recorded. Find the probability
Suppose that Y|X = x ∼ N(5 − 2x, 16), that is, given X = x, Y has the normal distribution with mean μY|X(x) = 5 − 2x and variance σ2ε = 16, and let σX = 3.(a) Let Y1 and Y2 be observations to be taken, independently from each other, when X has been observed to take the value 1 and 2,
Suppose that Y|X = x ∼ N(5 − 2x, 16), that is, given X = x, Y has the normal distribution with mean μY|X(x) = 5 − 2x and variance σ2ε = 16, and let σX = 3.(a) Find σ2Y and ρX,Y.(b) If Y1 is an observation to be taken when X has been observed to take the value 1, find the 95th percentile
The bivariate normal distribution. X and Y are said to have a bivariate normal distribution if their joint distribution is specified according to the hierarchical modelGive an expression of the joint PDF of X and Y. (Bo Y|X =x^N +B(x - "x), ) 2 and X ~N(x,). (4.6.2)
Let X be the number of eggs an insect lays and Y the number of eggs that survive.Suppose each egg survives with probability p, independently of other eggs.Use the principle of hierarchical modeling to describe a reasonable model for the joint distribution of X and Y.
9. Let X be defined by the probability density function(a) Define Y = X2 and find Cov(X,Y).(b) Without doing any calculations, find the regression function E(Y|X = x) (Hint. When the value of X is given, the value of Y is known).(c) On the basis of the regression function found above, comment on
8. Let X have the uniform in (−1, 1) distribution and let Y = X2. Using calculations similar to those in Example 4.5-3, show that ρX,Y = 0.
7. Consider the context of Exercise 17 in Section 4.3, so that the variables X = static force at failure, Y = defect index, have joint PDF(a) 9It is given that the marginal density of X is fX(x) = 1−2x 0 24x dy = 24x(1 − 2x), 0 ≤ x ≤ 0.5, and the marginal density of Y is fY(y) = 9 (1−y)/2
6. Select two products from a batch of 10 containing three defective and seven non-defective products. Let X = 1 or 0 as the first selection from the 10 products is defective or not, and Y = 1 or 0 as the second selection (fromthe nine remaining products) is defective or not.(a) Find the marginal
5. Import the bear data into R with the R command br=read.table(”BearsData.txt”, header=T), and form a data frame consisting only of the measurements with the R commands attach(br); bd=data.frame(Head.L, Head.W, Neck.G, Chest.G, Weight).3 The R command cor(bd)returns a matrix of the pairwise
4. Use ta=read.table(”TreeAgeDiamSugarMaple.txt”, header=T) to import the data set of diameter–age measurements for 27 sugar maple trees into the R data frame ta, and x=ta$Diamet; y=ta$Age to copy the diameter and age values into the R objects x and y, respectively.(a) Would you expect the
3. An article2 reports data on X = distance between a cyclist and the roadway center line and Y = the separation distance between the cyclist and a passing car, from ten streets with bike lanes. The paired distances (Xi,Yi)are determined by photography and are given below in feet.(a) Compute SX,Y,
2. Consider the information given in Exercise 2 in Section 4.2 regarding the amount, X, of drug administered to a randomly selected laboratory rat, and number, Y, of tumors the rat develops.(a) Would you expect X and Y to be positively or negatively correlated? Explain your answer, and confirm it
1. Using the joint distribution, given in Exercise 1 in Section 4.3, of the volume of monthly book sales from the online site, X, and the volume of monthly book sales from the brick and mortar counterpart, Y, of a bookstore, compute the linear correlation coefficient of X and Y.
To calibrate a method for measuring lead concentration in water, the method was applied to 12 water samples with known lead content. The concentration measurements, y, and the known concentration levels, x, are given belowCompute the sample covariance and correlation coefficient. x 5.95 2.06 1.02
Let X have the uniform in (0, 1) distribution, and Y = X2. Find ρX,Y.
Using a geolocation system, a dispatcher sends messages to two trucks sequentially.Suppose the joint PDF of the response times X1 and X2, measured in seconds, is f (x1, x2) = exp(−x2) for 0 ≤ x1 ≤ x2, and f (x1, x2) = 0 otherwise.(a) Find Corr(X1,X2).(b) If ((X1, (X2) denote the response
In a reliability context a randomly selected electronic component will undergo an accelerated failure time test. Let X1 take the value 1 if the component lasts less than 50 hours and zero otherwise, and X2 take the value 1 if the component lasts between 50 and 90 hours and zero otherwise. The
16. Let X have the negative binomial distribution with parameters r and p. Thus, X counts the total number of Bernoulli trials until the rth success. Next, let X1 denote the number of trials up to and including the first success, let X2 denote the number from the first success up to and including
15. Let X be a hypergeometric random variable with parameters n, M1, and M2. Use Corollary 4.4-1 to give an alternative (easier) derivation of the formula for E(X).(Hint. See the derivation of the expected value of a Bin(n, p) random variable following Corollary 4.4-1.)
14. On the first day of a wine-tasting event three randomly selected judges are to taste and rate a particular wine before tasting any other wine. On the second day the same three judges are to taste and rate the wine after tasting other wines. Let X1,X2,X3 be the ratings, on a 100-point scale, on
13. Let X, Y, and Z be independent uniform(0, 1) random variables, and set X1 = X + Z, Y1 = Y + 2Z.(a) Find Var(X1 + Y1) and Var(X1 − Y1). (Hint. Find Var(X1), Var(Y1), Cov(X1,Y1), and use part (1) of Proposition 4.4-4.)(b) Using R commands similar to those used in Example 4.4-7, generate a
12. Consider the information given in Exercise 7 in Section 4.3 regarding the level of moisture content and impurity of chemical batches. Such batches are used to prepare a particular substance. The cost of preparing the substance is C = 2√X+3Y2. Find the expected value and variance of the cost
11. Consider the information given in Exercise 1 in Section 4.3 on the joint distribution of the volume, X, of online monthly book sales, and the volume, Y, of monthly book sales from the brick and mortar counterpart of a bookstore. An approximate formula for the monthly profit, in thousands of
10. Using the information on the joint distribution of meal price and tip given in Exercise 3 in Section 4.3, find the expected value and the variance of the total cost of the meal (entree plus tip) for a randomly selected customer.
9. Suppose the random variables Y, X, and ϵ are related through the model Y = 9.3 + 1.5X + ε, where ε has zero mean and variance σ2ε = 16, σ2X= 9, and X, ε are independent. Find the covariance of Y and X and that of Y and ε. (Hint. Write Cov(X,Y) = Cov(X, 9.3 + 1.5X + ε) and use part (4)
8. Suppose (X,Y) have the joint PDFFind Cov(X,Y). (Hint. Use the marginal PDF of X, which was derived in Example 4.3-9, and note that by the symmetry of the joint PDF in x, y, it follows that the marginal PDF of Y is the same as that of X.) f(x,y) = 24xy 0x1,0 y 1,x+y1 otherwise.
7. In a typical evening, a waiter serves N1 tables that order alcoholic beverages and N2 tables that do not.Suppose N1, N2 are Poisson random variables with parameters λ1 = 4, λ2 = 6, respectively. Suppose the tips, Xi, left at tables that order alcoholic beverages have common mean value of
6. Let N denote the number of accidents per month in all locations of an industrial complex, and let Xi denote the number of injuries reported for the ith accident. Suppose that the Xi are independent random variables having common expected value of 1.5 and are independent from N. If E(N) = 7, find
5. Two towers are constructed, each by stacking 30 segments of concrete vertically. The height (in inches) of a randomly selected segment is uniformly distributed in the interval (35.5, 36.5).(a) Find the mean value and the variance of the height of a randomly selected segment. (Hint. See Examples
4. In a typical week a person takes the bus five times in the morning and three times in the evening. Suppose the waiting time for the bus in the morning has mean 3 minutes and variance 2 minutes2, while the waiting time in the evening has mean 6 minutes and variance 4 minutes2.(a) Let Xi denote
3. The joint distribution of X = height and Y = radius of a cylinder is f (x, y) = 3x/(8y2) for 1 ≤ x ≤ 3, 0.5 ≤ y ≤ 0.75 and zero otherwise. Find the variance of the volume of a randomly selected cylinder. (Hint. The volume of the cylinder is given by h(X,Y) = πY2X. In Example 4.3-17 it
2. A system consists of components A and B connected in parallel. Suppose the two components fail independently, and the time to failure for each component is a uniform(0, 1) random variable.(a) Find the PDF of the time to failure of the system.(Hint. If X, Y are the times components A, B fail,
1. Due to promotional sales, an item is sold at 10% or 20% below its regular price of $150. Let X and Y denote the selling prices of the item at two online sites, and let their joint PMF beIf a person checks both sites and buys from the one listing the lower price, find the expected value and
If X,Y have the joint PMF given byfind Cov(X,Y). Are X and Y independent? y p(x,y) 0 1 -1 1/3 0 1/3 X 0 0 1/3 1/3 1 1/3 0 1/3 2/3 1/3 1.0
Consider the information given in Example 4.4-9, but assume that the response times are given in milliseconds. If ((X1, (X2) denote the response times inmilliseconds, find Cov((X1, (X2).
Using a geolocation system, a dispatcher sends messages to two trucks sequentially.Suppose the joint PDF of the response times X1 and X2, measured in seconds, is f (x1, x2) = exp(−x2) for 0 ≤ x1 ≤ x2, and f (x1, x2) = 0 otherwise. Find Cov(X1,X2).
Let X take the value 0, 1, or 2 depending on whether there are no customers, between one and 10 customers, and more than 10 customers in the regular (manned)checkout lines of a supermarket. Let Y be the corresponding variable for the self checkout lines. Find Var(X + Y), when the joint PMF of X and
Simulation-based verification of part (1a) of Proposition 4.4-4. Let X, Y be independent uniform(0, 1) random variables. Generate a random sample of size 10,000 of X+Y values, and a random sample of size 10,000 of X−Y values and compute the sample variances of the two samples. Argue that this
Let N denote the number of people entering a department store in a typical day, and let Xi denote the amount of money spent by the ith person. Suppose the Xi have a common mean of $22.00, independently from the total number of customers N. If N is a Poisson random variable with parameter λ = 140,
In a typical evening, a waiter serves four tables that order alcoholic beverages and three that do not.(a) The tip left at a table that orders alcoholic beverages is a random variable with mean μ1 = 20 dollars. Find the expected value of the total amount of tips the waiter will receive from the
If X and Y are independent, show that, for any functions g and h, E[g(X)h(Y)]E[g(X)]E[h(Y)].
A system consists of components A and B connected in series. If the two components fail independently, and the time to failure for each component is a uniform(0, 1)random variable, find the expected value and variance of the time to failure of the system.
A photo processing website receives compressed files of images with X × Y pixels where X and Y are random variables. At compression factor 10:1, 24 bits-per-pixel images result in compressed images of Z = 2.4XY bits. Find the expected value and variance of Z when the joint PMF of X and Y is y
18. Consider the context of Exercise 17.(a) 9It is given that the marginal density of X is fX(x) = 1−2x 0 24x dy = 24x(1 − 2x), 0 ≤ x ≤ 0.5. Find fY|X=x(y) and the regression function E(Y|X = x).Plot the regression function and give the numerical value of E(Y|X = 0.3).(b) Use the Law of
17. A type of steel has microscopic defects that are classified on a continuous scale from 0 to 1, with 0 the least severe and 1 the most severe. This is called the defect index. Let X and Y be the static force at failure and the defect index, respectively, for a particular type of structural
16. Let X be the force (in hundreds of pounds) applied to a randomly selected beam and Y the time to failure of the beam. Suppose that the PDF of X is fX(x) =1 log(6) − log(5)1 xfor 5 ≤ x ≤ 6 and zero otherwise, and that the conditional distribution of Y given that a force X = x is applied is
15. Let X and Y have the joint PDF of Example 4.3-5.Use the form of the conditional PDF of X given Y = y for y > 0, derived there, to conclude whether or not X and Y are independent. (Hint. Use part (4) of Proposition 4.3-2.)
14. During a typical Pennsylvania winter, potholes along I80 occur according to a Poisson process averaging 1.6 per 10 miles.Acertain county is responsible for repairing potholes in a 30-mile stretch of I80. At the end of winter the repair crew starts inspecting for potholes from one end of the
13. Let Ti, i = 1, 2, denote the first two interarrival times of a Poisson process X(s), s ≥ 0, with rate α. (So, according to Proposition 3.5-1, both T1 and T2 have an exponential distribution with PDF f (t) = αe−αt, t > 0.)Show that T1 and T2 are independent. (Hint. Argue that P(T2 > t|T1
12. Criterion for independence. X and Y are independent if and only iffor some functions g and h (which need not be PDFs).[An important point to keep in mind when applying this criterion is that condition (4.3.20) implies that the region of (x, y) valueswhere f (x, y) is positive has to be a
11. The joint PDF of X and Y is f (x, y) = x + y for 0
10. It is known that, with probability 0.6, a new laptop owner will install a wireless Internet connection at home within a month. Let X denote the number (in hundreds)of new laptop owners in a week from a certain region, and let Y denote the number among them who install a wireless connection at
9. Let X be the force applied to a randomly selected beam for 150 hours, and let Y take the value 1 or 0 depending on whether the beam fails or not. The random variable X takes the values 4, 5, and 6 (in 100-lb units)with probability 0.3, 0.5, and 0.2, respectively. Suppose that the probability of
8. Consider the information given in Exercise 7.(a) Find the regression function, μY|X(x), of Y on X.(b) Use the Law of Total Expectation to find E(Y).
7. 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 of a randomly selected batch, respectively. Use the information given in
6. Consider the information given in Exercise 5.(a) Find the regression function, μY|X(x), of Y on X.(b) Use the Law of Total Expectation to find E(Y).
5. Let X take the value 0 if a child between 4 and 5 years of age uses no seat belt, 1 if he or she uses a seat belt, and 2 if it uses a child seat for short-distance car commutes.Also, let Y take the value 0 if a child survived a motor vehicle accident and 1 if he or she did not. Accident records
4. Consider the information given in Exercise 2 in Section 4.2.(a) What is the conditional PMF of the number of tumors for a randomly selected rat in the 1.0 mg/kg drug dosage group?(b) Find the regression function of Y, the number of tumors present on a randomly selected laboratory rat, on X, the
3. Let X, Y be as in Exercise 3 in Section 4.2.(a) Find the regression function Y on X.(b) Use the Law of Total Expectation to find E(Y).(c) Is the amount of tip left independent of the price of the meal? Justify your answer in terms of the regression function. (Hint. Use part (1) of Proposition
2. Let X, Y have the joint PMF given in Exercise 1.(a) Find the regression function Y on X.(b) Use the Law of Total Expectation to find E(Y).
1. Let X denote the monthly volume of book sales from the online site of a bookstore, and let Y denote the monthly volume of book sales from its brick and mortar counterpart. The possible values of X and Y are 0, 1, or 2, in which 0 represents a volume that is below expectations, 1 represents a
Let the height, X, and radius, Y, both measured in centimeters, of a cylinder randomly selected from the production line have the joint PDF given in Example 4.3-15.(a) Find the expected volume of a randomly selected cylinder.(b) Let X1, Y1 be the height and radius of the cylinder expressed in
Consider the two-component system described in Example 4.3-14, and suppose that the failure of component A incurs a cost of $500.00, while the failure of component B incurs a cost of $750.00. Let CA and CB be the costs incurred by the failures of components A and B, respectively. Are CA and CB
For a cylinder selected at random from the production line, let X be the cylinder’s height and Y the cylinder’s radius. Suppose X,Y have a joint PDFAre X and Y independent? 38 f(x, y) = 8y2 0 if 1 x3, v= otherwise.
A system is made up of two components, A and B, connected in parallel. Let X take the value 1 or 0 if component A works or not, and Y take the value 1 or 0 if component B works or not. From the repair history of the system it is known that the conditional PMFs of Y given X = 0 and X = 1 areAre X
Consider the joint distribution of the two types of errors, X and Y, a robot makes, as given in Example 4.3-1. Are X,Y independent?
Let Y denote the age of a tree, and let X denote the tree’s diameter at breast height.Suppose that, for a particular type of tree, the regression function of Y on X isμY|X(x) = 5 + 0.33x and that the average diameter of such trees in a given forested area is 45 cm. Find the mean age of this type
Use the regression function of Y on X, and themarginal PDF of X,which were found in Example 4.3-9, in order to find E(Y). E(Y|X=x)=(1-x) and fx(x) = 12x(1-x), 0x1,
Use the regression function of Y on X and the marginal PMF of X,which were given in Examples 4.3-8 and 4.3-4, respectively, in order to find E(Y). X 0 1 2 x 0 1 2 and HYX(X) 0.2 0.95 1.25 Px(x) 0.2 0.5 0.3
Suppose (X,Y) have joint PDFFind the regression function of Y on X. 24xy 0x1,0 y 1,x+y1 f(x,y) = otherwise.
Let X be the force (in hundreds of pounds) applied to a randomly selected beam and Y the time to failure of the beam. Suppose that the PDF of X isand zero otherwise, and that the conditional distribution of Y, given that a force X = x is applied, is exponential(λ = x). Thus,and fY|X=x(y) = 0 for y
Let X, Y have the joint PDF given in Example 4.3-5.(a) Find P(X > 1|Y = 3).(b) Find the conditional mean and variance of X given that Y = 3.
The joint PDF of X and Y is f (x, y) = 0 if either x or y is Find fX|Y=y(x). e-x/ye-y f(x,y) = for x > 0, y > 0. y
Let X take the value 0, 1, or 2 depending on whether there are no customers, between 1 and 10 customers, and more than 10 customers in the regular (manned)checkout lines of a supermarket. Let Y be the corresponding variable for the self checkout lines. An extensive study undertaken by the
Let X and Y be as in Example 4.3-1. The conditional PMF of Y given X = 0 was found there to beCalculate the conditional expected value and variance of Y given that X = 0. y 0 1 2 3 PYX-0(y) 0.9333 0.0333 0.0222 0.0111
Let X(t) be a Poisson process with rate α. Find the conditional PMF of X(0.6) given X(1) = n (i.e., given that there are n occurrences in the time period [0, 1]).
Arobot performs two tasks,welding joints and tightening bolts. Let X be the number of defective welds and Y be the number of improperly tightened bolts per car. The joint and marginal PMFs of X and Y are given in the table below:Find the conditional PMF of Y given X = 0. y p(x,y) 0 1 2 3 Px(x) 0
8. Let the random variables X and Y have the joint PDF given below:(a) Find P(X + Y ≤ 3).(b) Find the marginal PDFs of Y and X. 2e-0x y < otherwise. f(x,y) = 0
7. Let the random variables X and Y have the joint PDF given below:f (x, y) = kxy2 for 0 ≤ x ≤ 2, x ≤ y ≤ 3.(a) Find the constant k. (Hint. Use the property that the volume under the entire surface defined by f (x, y) is 1.)(b) Find the joint CDF of X and Y.
6. When being tested, an integrated circuit (IC) is considered as a black box that performs certain designed functions. Four ICs will be randomly selected from a shipment of 15 and will be tested for static voltages, external components associated with the IC, and dynamic operation. Let X1, X2, and
5. Let X1, X2, and X3 denote the number of customers in line for self checkout, for regular checkout, and for express (15 items of less) checkout, respectively. Let the values 0, 1, and 2 denote zero customers, one customer, and two or more customers, respectively. Suppose the joint PMF, p(x1, x2,
4. The joint cumulative distribution function, or joint CDF, of the random variables X and Y is defined as F(x, y) = P(X ≤ x,Y ≤ y). Let X and Y be the random variables of Exercise 1.(a) Make a table for the F(x, y) at the possible (x, y)values that (X,Y) takes.(b) The marginal CDFs of X and Y
3. A local diner offers entrees in three prices, $8.00,$10.00, and $12.00. Diner customers are known to tip either $1.50, $2.00, or $2.50 per meal. Let X denote the price of the meal ordered, and Y denote the tip left, by a random customer. The joint PMF of X and Y is(a) Find P(X ≤ 10,Y ≤ 2)
2. The joint PMF of X, the amount of drug administered to a randomly selected laboratory rat, and Y, the number of tumors the rat develops, is(a) Find the marginal PMF of X and that of Y.(b) What is the probability that a randomly selected rat has (i) one tumor, (ii) at least one tumor?(c) Given
1. Let X be the number of daily purchases of a luxury item from a factory outlet location and Y be the daily number of purchases made online. Let the values 1, 2, and 3 denote the number of purchases less than five, at least five but less than 15, and 15 or more, respectively.Suppose the joint PMF
Let X1,X2,X3 have the joint PDF given byand f (x1, x2, x3) = 0 if one or more of the xi is negative.(a) Find an expression for P(X1 ≤ t1,X2 ≤ t2).(b) Find FX1 (t1), the marginal CDF of X1.(c) Find fX1 (t1), the marginal PDF of X1. f(x1,x2, x3)=e1e2e3 for X1 > 0,x2 > 0, x3 > 0,
Consider the bivariate density function(a) Find the probability that X > Y.(b) Find the probability that X ≤ 0.6 and Y ≤ 0.4.(c) Find the marginal PDF of X and Y. 12(x + xy) f(x,y) = 0 0 x, y 1 otherwise.
In a batch of 12 laser diodes, three have efficiency below 0.28, four have efficiency between 0.28 and 0.35, and five have efficiency above 0.35. Three diodes are selected at random and without replacement. Let X1, X2, and X3 denote, respectively, the number of diodes with efficiency below 0.28,
Let X,Y have the joint PMF as shown in the following table.This PMF is illustrated in Figure 4-1.(a) Find P(0.5 (b) Find the marginal PMF of Y. X y p(x,y) 1 2 1 0.034 0.134 23 0.066 0.266 3 0.100 0.400
14. A randomvariable T is said to have a Weibull distribution with shape parameter α > 0 and scale parameterβ > 0 if its PDF is zero for t The CDF of aWeibull(α, β) distribution has the following closed form expression:When α = 1 the Weibull PDF reduces to the exponential PDF with λ =
13. A random variable T has a gamma distribution with shape parameter α > 0 and scale parameter β > 0 if its PDF is zero for negative values andwhere - is the gamma function defined by -(α) = ∞0 tα−1e−tdt. The most useful properties of the gamma function are: -(1/2) = π1/2, -(α) =
12. A random variable T is said to have the lognormal(μln, σln) distribution if logT ∼ N(μln, σ2 ln), where log is the natural logarithm. The mean value and variance of T areThe log-normal(0, 1) distribution is called the standard log-normal distribution.(a) Show that if T has the
11. Answer the following questions.(a) Use the R command x=runif(50) to generate a simulated sample of size 50 from the uniform(0, 1) distribution and use commands like those given in Section 3.5.2 to construct a normal Q-Q plot. Could the simulated sample of 50 have come from a normal
10. A machine manufactures tires with a tread thickness that is normally distributed with mean 10 millimeters(mm) and standard deviation 2 mm. The tire has a 50,000-mile warranty. In order to last for 50,000 miles the tread thickness must be at least 7.9 mm. If the thickness of tread is measured to
9. The finished inside diameter of a piston ring is normally distributed with a mean of 10 cm and a standard deviation of 0.03 cm.(a) Above what value of inside diameter will 85.08% of the piston rings fall?(b) What is the probability that the diameter of a randomly selected piston will be less
8. Admission officers in Colleges A and B use SAT scores as their admission criteria. SAT scores are normally distributed with mean 500 and standard deviation 80. College A accepts people whose scores are above 600, and College B accepts the top 1% of people in terms of their SAT scores.(a) What
7. The resistance for resistors of a certain type is a random variable X having the normal distribution with mean 9 ohms and standard deviation 0.4 ohms. A resistor is acceptable if its resistance is between 8.6 and 9.8 ohms.(a) What is the probability that a randomly chosen resistor is

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