Questions and Answers of Modern Mathematical Statistics With Applications

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
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?
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.
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
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
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 =
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
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
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
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
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
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
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
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
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
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
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
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
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
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?
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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 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
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.
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
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 =
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
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
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,
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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).
Explain why the transformation x = –μln(u) may be used to simulate values from an exponential distribution with mean μ.
An insurance company issues a policy covering losses up to 5 (in thousands of dollars). The loss, X, follows a distribution with density function ƒ(x) = 3/x4 for x ≥ 1 and = 0 otherwise. What
Let X have the pdf ƒ(x) = 1/[π(1 + x2)] for–∞ < x < ∞ (a central Cauchy distribution), and show that Y = 1/X has the same distribution.

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