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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 3rd Edition Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton - Solutions

It can be shown that if Yn converges in probability to a constant τ, then h(Yn) converges to h(τ) for any function h(·) that is continuous at τ. Use this to obtain a consistent estimator for the rate parameter k of an exponential distribution.
Apply the Law of Large Numbers to show that χv2 / v approaches 1 as v becomes large.
Show directly from the pdf that the mean of a t1 (Cauchy) random variable does not exist.
Show that the ratio of two independent standard normal random variables has the t1 distribution.
Derive the F pdf by applying the method used to derive the t pdf.
Let X have an F distribution with ν1 numerator df and ν2 denominator df.a. Determine the mean value of X.b. Determine the variance of X.
Is E(Fv1,v2) = E(χ2v1/ v1) / E(χ2v2 / v2)? Explain.
Show that Fp,v1,v2 = 1/F1–p,v2,v1 .
Suppose T ∼ t9. Determine the distribution of 1/T2.
Let Z1; Z2; X1; X2; X3 be independent rvs with each Zi ∼N(0,1) and each Xi ∼ N(0,5). Construct a variable involving the Zi’s and Xi’s which has an F3,2 distribution.
Let Z1, Z2, …, Z10 be independent standard normal. Use these to constructa. A χ24 random variable.b. A t4 random variable.c. An F4.6 random variable.d. A Cauchy random variable.e. An exponential random variable with mean 2.f. An exponential random variable with mean 1.g. A gamma random
The difference of two independent normal variables itself has a normal distribution. Is it true that the difference between two independent chi-squared variables has a chi-squared distribution? Explain.
Show that when sampling from a normal distribution, the sample variance S2 has a gamma distribution, and identify the parameters α and β.
Suppose X1; . . .; X27 are iid N(5,4) rvs. Let X̅ and S denote their sample mean and sample standard deviation, respectively. Calculate P(ΙX̅ – 5Ι > 0:4S).
A large university has 500 single employees who are covered by its dental plan. Suppose the number of claims filed during the next year by such an employee is a Poisson rv with mean value 2.3. Assuming that the number of claims filed by any such employee is independent of the number filed by any
If X ∼ Unif[0, 1], find a linear transformation Y = cX + d such that Y is uniformly distributed on [A, B], where A and B are any two numbers such that A < B. Is there any other solution? Explain.
a. If a measurement error X is uniformly distributed on [–1, 1], find the pdf of Y = |X|, which is the magnitude of the measurement error.b. If X ∼ Unif[–1, 1], find the pdf of Y = X2.c. If X ∼ Unif[–1, 3], find the pdf of Y = X2.
Consider the following pdf:where θ > 0 and τ > 0 are the parameters of the model. [This pdf is suggested for modeling waiting time in the article “A Model of Pedestrians’ Waiting Times for Street Crossings at Signalized Intersections” (Trans. Res. 2013: 17–28).]a. Write a function
The article “Determination of the MTF of Positive Photoresists Using the Monte Carlo Method” (Photographic Sci. Engr. 1983: 254–260) proposes the exponential distribution with parameter λ = .93 as a model for the distribution of a photon’s free path length (lm) under certain circumstances.
The Weibull distribution was introduced in Section 4.5.a. Find the inverse of the Weibull cdf.b. Write a program to simulate n values from a Weibull distribution. Your program should have three inputs: the desired number of simulated values n and the two parameters α and β. It should have a
In the previous exercise, suppose instead that Y is uniformly distributed on [0, 1]. Find the pdf of X = πY2. Geometrically speaking, why should X have a pdf that is unbounded near 0?
Let X ∼ Unif(0, 1). Determine the pdf of Y = tan [π(X – .5)]. The random variable Y has the Cauchy distribution, named after the famous mathematician.
Let X ∼ Unif(0, 1). Determine the pdf of Y = –ln(X).
If X is distributed as N(μ, σ), find the pdf of Y = eX. Verify that the distribution of Y matches the lognormal pdf provided in Section 3.5.
The plasma cholesterol level (mg/dL) for patients with no prior evidence of heart disease who experience chest pain is normally distributed with mean 200 and standard deviation 35. Consider randomly selecting an individual of this type. What is the probability that the plasma cholesterol levela. Is
Determine z∝ for the following:a. ∝ = .0055b. ∝ = .09c. ∝ = .663
If the pdf of a measurement error X isƒ(x) = .5e–|x|. –∞ < x < ∞ show that MX(t) 1/1 – t2 for |t|< 1.
Let X ∼ Unif[0, 1]. Find a linear function Y = g(X) such that the interval [0, 1] is transformed into [–5, 5]. Use the relationship for linear functions MaX + b(t) = ebtMX(at) to obtain the mgf of Y from the mgf of X. Compare your answer with the result of Exercise 32, and use this to obtain
The time (min) between successive visits to a particular website has pdf ƒ(x) = 4e−4x, x ≥ 0; ƒ(x) = 0 otherwise. Use integration by parts to obtain E(X) and V(X).
For the distribution of Exercise 14,a. Compute E(X) and σX.b. What is the probability that X is more than 1 standard deviation from its mean value?Data From Exercise 14 Let X denote the amount of space occupied by an article placed in a 1-ft³ packing container. The pdf of X is f(x) = 90x8 (1-x) 0
The article “Forecasting Postflight Hip Fracture Probability Using Probabilistic Modeling” (J. Biomech. Engr. 2019) examines the risk of bone breaks for astronauts returning from space, who typically lose density during missions. One quantity the article’s authors model is the midpoint
Reconsider the distribution of checkout duration X described in Exercises 1 and 11. Compute the following:a. E(X)b. V(X) and σXc. If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].Data From Exercise 1Data From Exercise 11 Let X
The time X (min) for a laboratory assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with A = 25 and B = 35.a. Write the pdf of X and sketch its graph.b. What is the probability that preparation time exceeds 33 min?c. What is the probability that
The grade point averages (GPAs) for graduating seniors at a college are distributed as a continuous rv X with pdfa. Sketch the graph of f(x).b. Find the value of k.c. Find the probability that a GPA exceeds 3.d. Find the probability that a GPA is within .25 of 3.e. Find the probability that a GPA
Let X denote the power (MW) generated by a wind turbine at a given wind speed. The article “An Investigation of Wind Power Density Distribution at Location With Low and High Wind Speeds Using Statistical Model” (Appl. Energy 2018: 442–451) proposes the Rayleigh distribution, with pdfas a
Recall the Coupon Collector’s Problem described in Exercise 106 of Chapter 2. Let X = the number of cereal boxes purchased in order to obtain all 10 coupons.a. Use a simulation program to estimate E(X) and SD(X). Also compute the estimated standard error of your answer. b. Repeat (a) with 20
Exercise 101 of Chapter 2 referred to a multiple-choice exam in which 10 of the questions have two options, 13 have three options, 13 have four options, and the other 4 have five options. Let X = the number of questions a student gets right, assuming s/he is completely guessing.a. Write a program
A derangement of the numbers 1 through n is a permutation of all n those numbers such that none of them is in the “right place.” For example, 34251 is a derangement of 1 through 5, but 24351 is not because 3 is in the 3rd position. We will use simulation to estimate the number of derangements
Suppose that 20% of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following
Individual A has a red die and B has a green die (both fair). If they each roll until they obtain five “doubles” (11, …, 66), what is the pmf of X = the total number of times a die is rolled? What are E(X) and SD(X)?
A shipment of 20 integrated circuits (ICs) arrives at an electronics manufacturing site. The site manager will randomly select 4 ICs and test them to see whether they are faulty. Unknown to the site manager, 5 of these 20 ICs are faulty.a. Suppose the shipment will be accepted if and only if none
A carnival game consists of spinning a wheel with 10 slots, nine red and one blue. If you land on the blue slot, you win a prize. Suppose your significant other really wants that prize, so you will play until you win.a. What is the probability you’ll win on the first spin?b. What is the
Refer back to the communication system of Example 3.51. Suppose a voice packet can be transmitted a maximum of 10 times; i.e., if the 10th attempt fails, no 11th attempt is made to re-transmit the voice packet. Let X = the number of times a message is transmitted. Assuming each transmission
A kinesiology professor, requiring volunteers for her study, approaches students one by one at a campus hub. She will continue until she acquires 40 volunteers. Suppose that 25% of students are willing to volunteer for the study, that the professor’s selections are random, and that the student
Newton’s generalization of the binomial theorem can be used to show that, for any positive integer r,Use this to derive the negative binomial mgf presented in this section. Then obtain the mean and variance of a negative binomial rv using this mgf. (1u): Σ(71') k=0 - uk
Consider the pmf given in Exercise 29 for the random variable Y = the number of moving violations for which the a randomly selected insured individual was cited during the last 3 years. Write a program to simulate this random variable, then use your simulation to estimate E(Y) and SD(Y). How do
Consider the pmf given in Exercise 31 for the random variable X = capacity of a purchased freezer. Write a program to simulate this random variable, then use your simulation to estimate both E(X) and SD(X). How do these compare to the exact values of E(X) and SD(X)?Data From Exercise 31An appliance
Suppose person after person is tested for the presence of a certain characteristic. The probability that any individual tests positive is .75. Let X = the number of people who must be tested to obtain five consecutive positive test results. Use simulation to estimate P(X ≤ 25).
Exercise 19 investigated Benford’s law, a discrete distribution with pmf given by p(x) = log10((x + 1)/x) for x = 1, 2, …, 9. Use the inverse cdf method to write a program that simulates the Benford’s law distribution. Then use your program to estimate the expected value and variance of this
The matching problem. Suppose that N items labeled 1, 2, …, N are shuffled so that they are in random order. Of interest is how many of these will be in their “correct” positions (e.g., item #5 situated at the 5th position in the sequence, etc.) after shuffling.a. Write program that simulates
Exercise 40 describes the game Plinko from The Price is Right. Each contestant drop between one and 5 chips down the Plinko board, depending on how well s/he prices several small items. Suppose the random variable C = number of chips earned by a contestant has the following distribution: The
Tickets for a particular flight are $250 apiece. The plane seats 120 passengers, but the airline will knowingly overbook (i.e., sell more than 120 tickets), because not every paid passenger shows up. Let t denote the number of tickets the airline sells for this flight, and assume the number of
Imagine the following simple game: flip a fair coin repeatedly, winning $1 for every head and losing $1 for every tail. Your net winnings will potentially oscillate between positive and negative numbers as play continues. How many times do you think net winnings will change signs in, say, 1000 coin
A small high school holds its graduation ceremony in the gym. Because of seating constraints, students are limited to a maximum of four tickets to graduation for family and friends. Suppose 30% of students want four tickets, 25% want three, 25% want two, 15% want one, and 5% want none.a. Write a
Let X be a rv with mean μ. Show that E(X2) ≥ μ2, and that E(X2) > μ2 unless X is a constant.
A small publisher employs two typesetters. The number of errors (in one book) made by the first typesetter has a Poisson distribution mean μ1, the number of errors made by the second typesetter has a Poisson distribution with mean μ2, and each typesetter works on the same number of books. Then if
A friend recently planned a camping trip. He had two flashlights, one that required a single 6-V battery and another that used two size-D batteries. He had previously packed two 6-V and four size-D batteries in his camper. Suppose the probability that any particular battery works is p and that
For a discrete rv X taking values in {0, 1, 2, 3, …}, we shall derive the following alternative formula for the mean:a. Suppose for now the range of X is {0, 1, …, N} for some positive integer N. By re-grouping terms, show thatb. Re-write each row in the above expression in terms of the cdf of
For a particular insurance policy the number of claims by a policy holder in 5 years is Poisson distributed. If the filing of one claim is four times as likely as the filing of two claims, find the expected number of claims.
For customers purchasing a full set of tires at a particular tire store, consider the eventsA = {tires purchased were made in the United States}B = {purchaser has tires balanced immediately}C = {purchaser requests front-end alignment}along with A′, B′, and C′. Assume the following
Suppose a store sells two different coffee makers of a particular brand, a basic model selling for $30 and a fancy one selling for $30 and a fancy one selling for $50. Let X denote the number of people among the next 25 purchasing this brand who choose the more expensive model. Then h(X) = revenue
A professional organization (for statisticians, of course) sells term life insurance and major medical insurance. Of those who have just life insurance, 70% will renew next year, and 80% of those with only a major medical policy will renew next year. However, 90% of policyholders who have both
Suppose a single gene controls the color of hamsters: black (B) is dominant and brown (b) is recessive. Hence, a hamster will be black unless its genotype is bb. Two hamsters, each with genotype Bb, mate and produce a single offspring. The laws of genetic recombination state that each parent is
The probability that a grader will make a marking error on any particular question of a multiple-choice exam is .1. If there are ten questions and questions are marked independently, what is the probability that no errors are made? That at least one error is made? If there are n questions and the
Professor Stander Deviation can take one of two routes on his way home from work. On the first route, there are four railroad crossings. The probability that he will be stopped by a train at any particular one of the crossings is .1, and trains operate independently at the four crossings. The other
It’s a commonly held misconception that if you play the lottery n times, and the probability of winning each time is 1/N, then your chance of winning at least once is n/N. That’s true if you buy n tickets in one week, but not if you buy a single ticket in each of n independent weeks. Let’s
Imagine you have five independently a operating components, each working properly with probability .8. Use simulation to estimate the probability thata. All five components work properly.b. At least one of the five components works properly.c. Calculate the estimated standard errors for your
Consider the system depicted in Exercise 89. Assume the seven components operate independently with the following probabilities of functioning properly: .9 for components 1 and 2; .8 for each of components 3, 4, 5, 6; and .95 for component 7. Write a program to estimate the reliability of the
You have an opportunity to answer six trivia questions about your favorite sports team, and you will win a pair of tickets to their next game if you can correctly answer at least three of the questions. Write a simulation program to estimate the chance you win the tickets under each of the
In the game “Now or Then” on the television show The Price is Right, the contestant faces a wheel with 6 sectors. Each sector contains a grocery item and a price, and the contestant must decide whether the price is “now” (i.e., the item’s price the day of the taping) or “then” (the
If you toss a fair die with outcome X, p(x) = 1/6 for x = 1, 2, 3, 4, 5, 6. Find MX(t).
If MX(t) = 1/(1 − t2), find E(X) and V(X) by differentiating MX(t).
Let X have the moment generating function of Example 3.29 and let Y = X − 1. Recall that X is the number of people who need to be checked to get someone who is Rh+, so Y is the number of people checked before the first Rh+ person is found. Find MY(t) using the last proposition in this
Show that g(t) = tet cannot be a moment generating function.
Use Appendix Table A.1 or software to obtain the following probabilities:a. B(4; 10, .3)b. b(4; 10, .3)c. b(6; 10, .7)d. P(2 ≤ X ≤ 4) when X ∼ Bin(10, .3)e. P(2 ≤ X) when X ∼ Bin(10, .3)f. P(X ≤ 1) when X ∼ Bin(10, .7)g. P(2 < X < 6) when X ∼ Bin(10, .3) a. n =
Let MX(t) be the moment generating function of a rv X, and define RX(t) = ln = [MX(t)]. Show thata. RX(0) = 0b. R'X(0) = μXc. R"X0 (0) = σ2X
If MX(t) = e5t + 2t2 then find E(X) and V(X) by differentiatinga. MX(t) b. RX(t)
If MX(t) = e5(et – 1) then find E(X) and V(X) by differentiatinga. MX(t) b. RX(t)
Let MX(t) = e5t + 2t2 and let Y = (X − 5)/2.Find MY(t) and use it to find E(Y) and V(Y).
a. Prove the result in the last proposition of this section: MaX +b(t) = ebtMX(at).b. Let Y = aX + b. Use (a) to establish the relationships between the means and variances of X and Y.
Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible. What is the probability that, of 20 randomly chosen drivers coming to an intersection under these conditions,a. At most 6 will come to a
A March 29, 2019, Washington Post article reported that (roughly) 5% of all students taking the ACT were granted extra time. Assume that 5% figure is exact, and consider a random sample of 25 students who have recently taken the ACT.a. What is the probability that exactly 1 was granted extra
Suppose that 90% of all batteries from a supplier have acceptable voltages. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at least
If X is a negative binomial rv, then the rv Y = X − r is the total number of failures preceding the rth success. a. Use an argument similar to the one presented in this section to derive the pmf of Y. b. Obtain the mgf of Y.c. Determine the mean and variance of Y. Are these intuitively
An electronics store has received a shipment of 20 table radios that have connections for an iPod or iPhone. Twelve of these have two slots (so they can accommodate both devices), and the other eight have a single slot. Suppose that six of the 20 radios are randomly selected to be stored under a
Let X, the number of flaws on the surface of a randomly selected carpet of a particular type, have a Poisson distribution with parameter μ = 5. Use software or Appendix Table A.2 to compute the following probabilities:a. P(X ≤ 8)b. P(X = 8)c. P(9 ≤ X)d. P(5 ≤ X ≤ 8)e. P(5 < X < 8) a.
Show that the binomial moment generating function converges to the Poisson moment generating function if we let n → ∞ and p → 0 in such a way that np approaches a value μ > 0. There is in fact a theorem saying that convergence of the mgf implies convergence of the probability
If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 400-page
Suppose that only .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers.a. What are the expected value and standard deviation of the number of computers in the sample that have the defect?b. What is the (approximate)
The article “Metal Hips Fail Faster, Raise Other Health Concerns” on the www. arthritis.com website reported that the five-year failure rate of metal-on-plasticimplants was 1.7% (rates for metal-onmetal and ceramic implants were significantly higher). Use both a binomial calculation and a
Bit transmission errors between computers sometimes occur, where one computer sends a 0 but the other computer receives a 1 (or vice versa). Because of this, the computer sending a message repeats each bit three times, so a 0 is sent as 000 and a 1 as 111. The receiving computer “decodes” each
A k-out-of-n system functions provided that at least k of the n components function. Consider independently operating components, each of which functions (for the needed duration) with probability .96. a. In a 3-component system, what is the probability that exactly two components function?b.
AAnn is expected at 7:00 pm after an all-day drive. She may be as much as one hour early or as much as three hours late. Assuming that her arrival time X is uniformly distributed over that interval, find the pdf of |X − 7|, the absolute difference between her actual and predicted arrival times.
If a measurement error X is distributed as N(0, 1), find the pdf of |X|, which is the magnitude of the measurement error.
If X ∼ Unif[0, 1], find a linear transformation Y = cX + d such that Y is uniformly distributed on [A, B], where A and B are any two numbers such that A < B. Is there any other solution? Explain.
a. If a measurement error X is uniformly distributed on [–1, 1], find the pdf of Y = |X|, which is the magnitude of the measurement error.b. If X ∼ Unif[–1, 1], find the pdf of Y = X2.c. If X ∼ Unif[–1, 3], find the pdf of Y = X2.
A circular target has radius 1 foot. Assume that you hit the target (we shall ignore misses) and that the probability of hitting any region of the target is proportional to the region’s area. If you hit the target at a distance Y from the center, then let X = πY2 be the corresponding area.
The amount of time (hours) required to complete an unusually short statistics homework assignment is modeled by the pdf ƒ(x) = x/2 for 0 < x < 2 (and = 0 otherwise).a. Obtain the cdf and then its inverse.b. Write a program to simulate 10,000 values from this distribution.c. Compare the
In distributed computing, any given task is split into smaller subtasks which are handled by separate subtasks which are handled by separate processors (which are then re-combined by a multiplexer). Consider a distributed computing system with 4 processors, and suppose for one particular purpose
Explain why the transformation x = –μln(u) may be used to simulate values from an exponential distribution with mean μ.

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