Tensor Methods In Statistics Monographs On Statistics And Applied Probability 1st Edition Peter McCullagh - Solutions

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1. Statistical Literacy Consider the following two events for an individual:A 5 owns a cell phone B 5 owns a laptop computer Translate each event into words.(a) Ac (d) A | B(b) A and B (e) B | A(c) A or B
2. Statistical Literacy If two events A and B are mutually exclusive, what is the value of P(A and B)?
3. Statistical Literacy If two events A and B are independent, how do the probabilities P(A) and P1A | B2 compare?
4. Interpretation You are considering two facial cosmetic surgeries. These are elective surgeries and their outcomes are independent. The probability of success for each surgery is 0.90. What is the probability of success for both surgeries? If the probability of success for both surgeries is less
5. Interpretation You are applying for two jobs, and you estimate the probability of getting an offer for the first job is 0.70 while the probability of getting an offer for the second job is 0.80. Assume the job offers are independent.(a) Compute the probability of getting offers for both jobs.
6. Critical Thinking You are given the information that P1A2 5 0.30 and P1B2 5 0.40.(a) Do you have enough information to compute P(A or B)? Explain.(b) If you know that events A and B are mutually exclusive, do you have enough information to compute P(A or B)? Explain.
7. Critical Thinking You are given the information that P1A2 5 0.30 and P1B2 5 0.40.(a) Do you have enough information to compute P(A and B)? Explain.(b) If you know that events A and B are independent, do you have enough information to compute P(A and B)? Explain.
8. Critical Thinking For a class activity, your group has been assigned the task of generating a quiz question that requires use of the formula for conditional probability to compute P1B | A2. Your group comes up with the following question: “If P1A and B2 5 0.40 and P1A2 5 0.20, what is the
9. Salary Raise: Women Does it pay to ask for a raise? A national survey of heads of households showed the percentage of those who asked for a raise and the percentage who got one (USA Today). According to the survey, of the women interviewed, 24% had asked for a raise, and of those women who had
10. Salary Raise: Men According to the same survey quoted in Problem 9, of the men interviewed, 20% had asked for a raise and 59% of the men who had asked for a raise received the raise. If a man is selected at random from the survey population of men, find the following probabilities: P(man asked
11. General: Thumbtack Drop a thumbtack and observe how it lands.(a) Describe how you could use a relative frequency to estimate the probability that a thumbtack will land with its flat side down.(b) What is the sample space of outcomes for the thumbtack?(c) How would you make a probability
12. Survey: Reaction to Poison Ivy Allergic reactions to poison ivy can be miserable.Plant oils cause the reaction. Researchers at Allergy Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure.A random sample of 1000 people with known allergies to poison
13. General: Two Dice In a game of craps, you roll two fair dice. Whether you win or lose depends on the sum of the numbers appearing on the tops of the dice. Let x be the random variable that represents the sum of the numbers on the tops of the dice.(a) What values can x take on?(b) What is the
14. Academic: Passing French Class records at Rockwood College indicate that a student selected at random has probability 0.77 of passing French 101. For the student who passes French 101, the probability is 0.90 that he or she will pass French 102. What is the probability that a student selected
15. Combination: City Council There is money to send two of eight city council members to a conference in Honolulu. All want to go, so they decide to choose the members to go to the conference by a random process. How many different combinations of two council members can be selected from the eight
16. Basic Computation Compute. (a) P7,2 (b) C7,2 (c) P3,3 (d) C4,4
17. Counting: Exam Answers There are five multiple-choice questions on an exam, each with four possible answers. Determine the number of possible answer sequences for the five questions. Only one of the sets can contain all five correct answers. If you are guessing, so that you are as likely to
18. Scheduling: College Courses A student must satisfy the literature, social science, and philosophy requirements this semester. There are four literature courses to select from, three social science courses, and two philosophy courses. Make a tree diagram showing all the possible sequences of
19. General: Combination Lock To open a combination lock, you turn the dial to the right and stop at a number; then you turn it to the left and stop at a second number. Finally, you turn the dial back to the right and stop at a third number.If you used the correct sequence of numbers, the lock
20. General: Combination Lock You have a combination lock. Again, to open it you turn the dial to the right and stop at a first number; then you turn it to the left and stop at a second number. Finally, you turn the dial to the right and stop at a third number. Suppose you remember that the three
1. Look at Figure 4-11, “Who’s Cracking the Books?”(a) Does the figure show the probability distribution of grade records for male students? for female students? Describe all the grade-record probability distributions shown in Figure 4-11. Find the probability that a male student selected at
1. Discuss the following concepts and give examples from everyday life in which you might encounter each concept. Hint: For instance, consider the “experiment”of arriving for class. Some possible outcomes are not arriving (that is, missing class), arriving on time, and arriving late.(a) Sample
2. Discuss the concepts of mutually exclusive events and independent events. List several examples of each type of event from everyday life.(a) If A and B are mutually exclusive events, does it follow that A and B cannot be independent events? Give an example to demonstrate your answer.Hint:
3. Although we learn a good deal about probability in this course, the main emphasis is on statistics. Write a few paragraphs in which you talk about the distinction between probability and statistics. In what types of problems would probability be the main tool? In what types of problems would
1. Statistical Literacy What are the requirements for a probability distribution?
2. Statistical Literacy List the criteria for a binomial experiment. What does the random variable of a binomial experiment measure?
3. Critical Thinking For a binomial probability distribution, it is unusual for the number of successes to be less than m 2.5s or greater than m 1 2.5s.(a) For a binomial experiment with 10 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five
4. Critical Thinking Consider a binomial experiment. If the number of trials is increased, what happens to the expected value? to the standard deviation? Explain.
5. Probability Distribution: Auto Leases Consumer Banker Association released a report showing the lengths of automobile leases for new automobiles. The results are as follows.(a) Use the midpoint of each class, and call the midpoint of the last class 66.5 months, for purposes of computing the
6. Ecology: Predator and Prey Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. In the National Park Service Scientific Monograph Series 11, Wolf Ecology and Prey Relationships on Isle Royale, Peterson gives results of many wolf–moose studies.
7. Insurance: Auto State Farm Insurance studies show that in Colorado, 55%of the auto insurance claims submitted for property damage are submitted by males under 25 years of age. Suppose 10 property damage claims involving automobiles are selected at random.(a) Let r be the number of claims made by
8. Quality Control: Pens A stationery store has decided to accept a large shipment of ball-point pens if an inspection of 20 randomly selected pens yields no more than two defective pens.(a) Find the probability that this shipment is accepted if 5% of the total shipment is defective.(b) Find the
9. Criminal Justice: Inmates According to Harper’s Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 16 federal inmates is selected.(a) What is the probability that 12 or more are serving time for drug dealing?(b) What is the probability that 7 or fewer are
10. Airlines: On-Time Arrivals Consumer Reports rated airlines and found that 80% of the flights involved in the study arrived on time (that is, within 15 minutes of scheduled arrival time). Assuming that the on-time arrival rate is representative of the entire commercial airline industry, consider
11. Agriculture: Grapefruit It is estimated that 75% of a grapefruit crop is good;the other 25% have rotten centers that cannot be detected until the grapefruit are cut open. The grapefruit are sold in sacks of 10. Let r be the number of good grapefruit in a sack.(a) Make a histogram of the
12. Restaurants: Reservations The Orchard Café has found that about 5% of the diners who make reservations don’t show up. If 82 reservations have been made, how many diners can be expected to show up? Find the standard deviation of this distribution.
13. College Life: Student Government The student government claims that 85%of all students favor an increase in student fees to buy indoor potted plants for the classrooms. A random sample of 12 students produced 2 in favor of the project. What is the probability that 2 or fewer in the sample will
14. Quota Problem: Financial Suppose you are a (junk) bond broker who buys only bonds that have a 50% chance of default. You want a portfolio with at least five bonds that do not default. You can dispose of the other bonds in the portfolio with no great loss. How many such bonds should you buy if
15. Theater: Coughs A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters especially, the irritation level rises with each muffled explosion. According to Dr. Brian Carlin, a Pittsburgh pulmonologist, in any large audience you’ll hear about 11 coughs
16. Accident Rate: Small Planes Flying over the western states with mountainous terrain in a small aircraft is 40% riskier than flying over similar distances in flatter portions of the nation, according to a General Accounting Office study completed in response to a congressional request. The
17. Banking: Loan Defaults Records over the past year show that 1 out of 350 loans made by Mammon Bank have defaulted. Find the probability that 2 or more out of 300 loans will default. Hint: Is it appropriate to use the Poisson approximation to the binomial distribution?
18. Car Theft: Hawaii In Hawaii, the rate of motor vehicle theft is 551 thefts per 100,000 vehicles (Reference: U.S. Department of Justice, Federal Bureau of Investigation). A large parking structure in Honolulu has issued 482 parking permits.(a) What is the probability that none of the vehicles
19. General: Coin Flip An experiment consists of tossing a coin a specified number of times and recording the outcomes.(a) What is the probability that the first head will occur on the second trial?Does this probability change if we toss the coin three times? What if we toss the coin four times?
20. Testing: CPA Exam Cathy is planning to take the Certified Public Accountant Examination (CPA exam). Records kept by the college of business from which she graduated indicate that 83% of the students who graduated pass the CPA exam. Assume that the exam is changed each time it is given. Let n 5
Describe the usual partial order on the set of partitions of the number k.Explain why this set, together with the usual partial order, does not form a lattice for k ≥ 5. Verify by direct inspection that the structure is a lattice for each k ≤ 4.
Show that, if complex-valued random variables are permitted, there are no restrictions on the moment spaces or on the cumulant spaces such as those discussed in Section 3.8.
Show that achievement of the limit ρ4 = ρ2 23 − p − 1 implies the following constraint on the third cumulantsρ2 23 − ρ2 13 − p + 1 = 0.
By counting the number of distinguishable ways of placing r identical objects in p+1 labelled boxes, show that, in p dimensions, rank (M ′r) ≤ ( ), where M′r is defined in Section 3.8.
Show that there exists a unique distribution whose odd cumulants are zero and whose even cumulants are κ2 = 1, κ4 = −2, κ6 = 16, κ8 = −272, κ10 =7936,…. Let M′r(λ) be the moment matrix described in Section 3.8, corresponding to the cumulant sequence, λκ1, …, λκ2r. Show that for
By considering the distribution of the sum of two independent random variables, show that the cumulant space, Kn is closed under vector addition.
By considering the mixture density, pf1(x) + (1 − p)f2(x), show that the moment space, Mn is convex.
In the notation of the previous exercise, prove that all cycles in the graph ϒ ⊕ϒ* have even length. (A cycle is a path beginning and ending at the same vertex.)Such a graph is said to be even. Show that all even connected graphs have a unique representation as ϒ ⊕ ϒ*. Hence prove that the
An alternative way of representing a partition by means of a graph is to use p labelled edges instead of labelled vertices. In this form, the graph of ϒ* ={υ1,…,υν} comprises p labelled edges emanating from α unlabelled vertices, giving a total of α+p vertices of which p are ‘free’. If
Generalize the result of the previous exercise to ordered k-tuplets of partitions. Give a simple explanation for this result.
Show that the number of ordered pairs (ϒ1,ϒ2) satisfying ϒ1 ⋁ ϒ2 = ϒ* for some fixed partition ϒ*, is∑ϒ≤ϒ∗m (ϒ, ϒ∗)B 2|υ1| …B 2|υν|, where m(ϒ, ϒ*) is the Möbius function for the partition lattice, defined below(3.12). Hence prove that the total number of ordered
Using the result given in Exercise 3.19, show that the total number of ordered pairs of partitions (ϒ1, ϒ2) satisfying ϒ1 ⋁ ϒ2 = 1 is C(2)p = ∑ϒ(−1)ν−1(ν − 1)!B 2|υ1| …B 2|υν|where the partitions contain p elements and B 2r is the square of the rth Bell number. Deduce also
Show that the number of sub-partitions of ϒ* is given by∑ϒ≤ϒ∗1 = B|υ∗1| …B|υ∗α|.
Interpret the expression in Exercise 3.19 as the cumulant of order αcum (X|υ∗1|,…, X|υ∗α|)where X is defined in Exercise 3.18 and the superscripts denote powers. Simplify this result in the special case where ϒ* is a 2κ partition and compare with Exercise 3.6.
Let ϒ* = {υ*1,…,υ*α} be a partition of a set of p elements into α non-empty blocks of sizes ∣υ*1∣,…,∣υ*α∣. By using (3.16) or otherwise, show that the number of partitions complementary to ϒ* is∑ (−1)ν−1(ν − 1)!B|υ1| …B|υν|.
Let X be a scalar random variable whose distribution is Poisson with mean 1.Show that the cumulants of X of all orders are equal to 1. Hence show that the rth moment isμ′r = E (Xr) = Br where Br, the rth Bell number, is the number of partitions of a set of r elements.Hence derive a generating
By reversing the order of summation in (3.17), show that the generalized cumulant may be writtenκ (ϒ∗) = ∑Πκ (π1)…κ (πσ) ∑ (−1)ν−1(ν − 1)!.Hence, using the result of the previous exercise, prove the fundamental identity(3.3).
MacCullagh’s formula: Using the notation of the previous exercise, let X i =Y i + κi where κi are the components of a vector of length ρ > 0 in the direction ϵ.Show that(MacCullagh, 1855; Jeffreys & Jeffreys, 1956, Section 18.09). [The above expression gives the potential experienced at an
Let Y = Y 1,…,Y phave zero mean and covariance matrix κi,j. Show that the‘total variance’, σ2 = E(Y iY jδij), is invariant under orthonormal transformation of Y. For any given direction, ϵ, define Q4 = a0 + aiXi + aijXiXj/2! + aijkXiXjXk/3!+aijklXiXjXkXl/4!,κ2 (U) =
Consider the first-order autoregressive process Y0 = ϵ0 = 0, Yj = βYj−1 +ϵj, j= 1,…, n, where ∣β∣ < 1 and ϵj are independent N(0,1) random variables. Show that the log likelihood function has first derivative U = ∂l/∂β = T1 − βT2 where T1 =∑YjYj−1 and T2 = ∑Y 2j−1 with
Show that any polynomial expression in X 1,…,X p, say can be expressed as a homogeneous polynomial in X 0,X 1,…,X pQ4 =p∑ijkl=0 bijklX iX jX kX l/4!, where X 0 = 1. Give expressions for the coefficients b in terms of the as. Find E(Q4) and var(Q4) and express the results in terms of the as.
Let X be a scalar random variable with moments μr and cumulants κr in the notation of Section 2.5. Show that the rth moment of the polynomial Y = P (X) = a0 + a1X + a2X2 + …is given formally by P(d)MX(ξ)∣ξ=0 where d = d/dξ. Hence show that the moment generating function of Y is MY (ς) =
If X = X 1, …,X pare jointly normal with zero mean, show that the joint cumulant generating function, logE, exp(ξijY ij) of Y i,j = X iX jis KY (ξ) = −1 12 log |I − 2ξκ|. Hence derive the cumulant generating function for the Wishart distribution.
For any positive definite matrix A, define the matrix B = log(A) by A = exp (B) = I + B + B2/2! + … + Br/r! + …By inverting this series, or otherwise, show that log |A| = tr log (A), where ∣A∣ is the determinant of A.
Show that κr,sκt,uξrtξsu= tr{(κξ)2} where κξ is the usual matrix product.
If X = X 1,…, X pare jointly normal with zero mean and co-variance matrix κi,j of full rank, show that the rth order cumulant of Y =κi,jX iX jis 2 r−1 p(r − 1)!.Hence show that the cumulant generating function of Y is −1 2 p log (1 − 2ξ) and therefore that Y has the χ2 distribution on
If ϒ* is a 2 k partition (a partition of 2k elements into k blocks of 2 elements each), show that the number of 2 k partitions ϒ satisfying ϒ ⋁ ϒ* = 1, is 2 k−1(k −1)!.
Show, in the terminology of Exercise 3.4, that M is indecomposable if and only if ϒ ⋁ ϒ* = 1. (Leonov & Shiryaev, 1959; Brillinger, 1975, Section 2.3.)
Let M be the intersection matrix ϒ* ∩ ϒ. Columns j1 and j2 are said to hook if, for some i, mij1 > 0 and mij2 > 0. The set of columns is said to communicate if there exists a sequence j1, j2,…, jνsuch that columns jι and jι+1 hook. The matrix M is said to be indecomposable if its columns
Using the results derived in the previous exercise, find an expression for the variance of the residual sum of squares in the standard linear regression model.
Let X = X 1,…,X nbe independent and let the residuals X i − κi be identically distributed with cumulants κ2δij, κ3δijk and so on. Find the mean and variance of the quadratic form aijX iX j and express the result in matrix notation (Seber, 1977, p. 9).
Let X 1,…,X nbe independent and identically distributed. By expressing X̅ as a linear form and the sample variance, s 2as a quadratic form, show that cov(X̅, s 2) =κ3/n. Hence show that corr(X̅, s 2) → ρ3/(2 + ρ4)1/2 as n → ∞. Show also that cov(X̅2, s 2) = κ4/n + 2κ1κ3/n and show
Using the notation established in the previous exercise, show that Give an expression for the cumulant corresponding to an arbitrary partition of the indices.
More generally, using the notation of the previous exercise, show that, for i> j > κ,∫ hi (x)hj (x)hk (x)ϕ (x)dx = i!j!k!{1 2 (j+k−i)}!{1 2 (i+k−j)}!{1 2 (i+j−k)}!when j+k−i is even and non-negative, and zero otherwise, (Jarrett, 1973, p. 26).
Using (5.26) or otherwise, show that in the univariate case, where X is a standardized sum with mean zero, unit variance and so on, then Y∗ = X − ρ3X2/6 − ρ4X3/24 + ρ2 3X3/9 has mean − ρ3/6 and standard deviation 1 − ρ4/8 + 7ρ2 3/36 when terms of order O(n −3/2) are ignored. Show
Taking the definition of Y* as given in the previous exercise, show that W/2 = (Y∗)2 = X2/2 − ρ3X3/3! − {ρ4 − 3ρ2 3}X4/4!has mean given by E (W) = 1 + (5ρ2 3 − 3ρ4)/12 = 1 + b/n.Deduce that W1+b/n~χ2 1 + O (n−3/2).
Under the assumption that the data have a joint normal distribution, show that n 1/2 kr,s,t,u has a limiting covariance matrix given by n cov (k i,j,k,l, k r,s,t,u) → κi,rκj,sκk,tκl,u[4!].Hence show that n 1/2 pr̅4 has a limiting normal distribution with mean zero and variance 8p 2+ 16p.
Using the equation following (5.26), show by reversal of series, that X may be expressed as the following polynomial in the normally distributed random variable Y X = Y + ρ3 (Y 2 − 1)/6 + ρ4 (Y 3 − 3Y )/24 − ρ2 3 (2Y 3 − 5Y )/36 Hence, express the approximate percentage points of X in
Let X = Y + Z where Y has density function f0(y) and Z is independent of Y with moments ηi, ηij, ηijk,…. Show formally, that the density of X is given by fX (x) = EZ {f0 (x − Z)}.Hence derive the series (5.2) by Taylor expansion of f0(x)• By taking ηi = 0, ηi,j= 0, and f0(x) = ϕ(x;
Show that where hr(x) is the standard univariate Hermite polynomial of degree r and ϕ(x)is the standard normal density. [Hint: use the tables of connecting partitions.]
Generalize the result of the previous exercise by showing that the joint cumulant corresponding to an arbitrary set of Hermite tensors involves a sum over connecting partitions. Describe the rule that determines the contribution of each connecting partition.
Using the notation established in the previous exercise, show that
Suppose now that X has cumulants κr, κr,s, κr,s,t,… and that the Hermite tensors are based on the normal density with mean λr and covariance matrix λr,s.Show that and so on, where the ηs are defined in Section 5.2.1.
Show, under conditions to be stated, that if fX (x; κ) = f0 (x) + ηifi (x) + ηijfij (x)/2! + ηijkfijk (x)/3! + …then the moment generating function of fX(x;κ) is M0 (ξ) {1 + ξiηi + ξiξjηij/2! + ξiξjξkηijk/3! + …}where Μ0(ξ) is the moment generating function of f0(x).
Using expansion (5.2) for the density, derive expansion (5.4) for the log density.
Give a heuristic explanation for the formal similarity of expansions (5.2) and those in Section 5.2.3.
Show that any generalized Hermite tensor involving β indices partitioned into a blocks, is of degree β−2α − 2 in x or is identically zero if β − 2α − 2 < 0.
Show that the mode of a density that can be approximated by an Edgeworth series occurs at xˆi = −κi,j,kκj,k/2 + O (n−3/2).
Show that the median of a univariate density that can be approximated by an Edgeworth series occurs approximately at the point xˆ =−κ3 6κ2.Hence show that, to the same order of approximation, in the univariate case,(mean−median)(mean−mode) =1 3(Haldane, 1942). See also Haldane (1948) for a
If fX(x) is the density function of X 1,…,X p, show that the density of Yr = a r + a riXi is fY (y) = JfX {b ir (y r − a r)}, where b ira rj = δi j and J is the determinant of b ir. Hence show that the partial derivatives of log fX(x) are Cartesian tensors.
Show that kr,s = n−1∑ijϕijY ri Y sj is the usual sample covariance matrix based on n independent and identically distributed observations. Show that k r,s is unbiased for κr,s.
By applying the criteria of unbiasedness and symmetry to k r,s,t and kr,s,t,u, derive the expression (4.7) for the coefficients ϕijk and ϕijkl.
Show that, for the ordinary linear regression problem with an intercept and one dependent variable, x, that n22 = (n − 1)(n − 3)/n +∑(xi − x)4/(∑(xi − x)2)2 and that, for equally spaced x-values, this reduces to n22 = (n − 2)2/n + 4/ (5n) + O (n−2)in reasonable agreement with the
By expanding the product of several power sums along the lines suggested in Section 4.6.2 for products of k-statistics, show that K = HP in the notation of (4.20), where P is a vector of polykays, K is a vector of power sum products and H has components hij = n′ (l = |ϒi ∨ ϒj| − |ϒi|),
Show, in the notation of Section 4.7.1, that if the scalar n22 =ρi,j,k,lρi,jρk,l is non-zero, then n−1 22 ρi,j,k,lρk,lRr;iRs;j is an invariant unbiased estimate of κr,s Under what circumstances is the above estimate the same as k r,s?
Show that the covariance of k r,s and k t,u, as defined in cov (k r,s, k t,u) = n22k r,s,t,u/n 22 + k r,tk s,u[2]/n2

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Tensor Methods In Statistics Monographs On Statistics And Applied Probability 1st Edition Peter McCullagh - Solutions

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