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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions

=+27.18. 21.21 Stirling's formula. Let S ,, = X, + . . . +X ,,, where the X ,, are indepen-dent and each has the Poisson distribution with parameter 1. Prove succes-sively- n F-"E(") k ="n - k n" +(1/2)e -1(a) E In n!Sp -n(b)=N7.Vn S, - n 1(c) E-EN-]=Vn 12 TT(d) n! ~ 1/2" n" +(1/2) e **
=+27.19. Let ( ,, (w) be the length of the run of 0's starting at the nth place in the dyadic expansion of a point w drawn at random from the unit interval; see Example 4.1.
=+(a) Show that /1, /2 ,... is an a-mixing sequence, wherea, == 4/2".
=+(b) Show that ER _, 4, is approximately normally distributed with mean n and variance 6n.
=+27.20. Prove under the hypotheses of Theorem 27.4 that S ,, /n -+ 0 with probability 1.Hint: Use (27.25).
=+27.21. 26.1 26.291 Let X1, X2 ,... be independent and identically distributed, and suppose that the distribution common to the X ,, is supported by [0, 2 m] and is not a lattice distribution. Let S ,, - X, + . . . + X ,, where the sum is reduced modulo 2Tr. Show that S ,, - U, where U is
=+28.1. Show that u. , & implies u(R1) < liminf, je (R1). Thus in vague conver-gence mass can "escape to infinity" but mass cannot "enter from infinity."
=+28.2. (a) Show that u ,, -, u if and only if (28.2) holds for every continuous f with bounded support.
=+(b) Show that if u, -, but (28.1) does not hold, then there is a continuous f vanishing at +00 for which (28.2) does not hold.
=+28.3. 23.71 Suppose that N, Y ,, Y ,,... are independent, the Y ,, have a common distribution function F, and N has the Poisson distribution with meana. Then S =Y1 + . . . + YN has the compound Poisson distribution.
=+(a) Show that the distribution of S is infinitely divisible. Note that S may not have a mean.
=+(b) The distribution function of S is En_be "a"F"(x)/n !, where F"* is the n-fold convolution of F (a unit jump at 0 for n = 0). The characteristic function of S is exp a (" -_ (ex - 1) dF(x).
=+(c) Show that, if F has mean 0 and finite variance, then the canonical measure
=+# in (28.6) is specified by p( A) =afax2 dF(x).
=+28.4. (a) Let v be a finite measure, and define itx)+2 (de).(28.12) +(1) = expiyt+ | |ex - 1 - -1 +x2
=+where the integrand is - t2/2 at the origin. Show that this is the characteristic function of an infinitely divisible distribution.
=+(b) Show that the Cauchy distribution (see the table on p. 348) is the case where y =0 and v has density w (1 +x2)"! with respect to Lebesgue measure.
=+28.5. Show that the Cauchy, exponential, and gamma (see (20.47)) distributions are infinitely divisible.
=+28.6. Find the canonical representation (28.6) of the exponential distribution with mean 1:
=+(a) The characteristic function is free""dx =(1-i)-1 =((t).
=+(b) Show that (use the principal branch of the logarithm or else operate formally for the moment) d(log (t))/dt = io(t) = ifgeile"" dx. Integrate with respect to t to obtain-
=+(28.13)Jo ("x -1) , dx .1 - it exp Verify (28.13) after the fact by showing that the ratio of the two sides has derivative 0.
=+(c) Multiply (28.13) by e-" to center the exponential distribution at its mean:The canonical measure u has density xe -* over (0,00).
=+28.7. 1 If X and Y are independent and each has the exponential density e "", then X -Y has the double exponential density ge -IF (see the table on p. 348).Show that its characteristic function is I +3 =exp f"(e"x- 1-ix) /xle-M dr.
=+28.8. 1 Suppose X1, X2 ,... are independent and each has the double exponential density. Show that 27- X ,. /n converges with probability 1. Show that the distribution of the sum is infinitely divisible and that its canonical measure has density |x|e -* /(1-e-(x) =C_|x|e-[x]
=+28.9. 26.8 Show that for the gamma density e-"x"-1/T(u) the canonical mea-sure has density uxe "" over (0,00).
=+The remaining problems require the notion of a stable law. A distribution function F is stable if for each n there exist constants a ,, and b ,.,a, > 0, such that, if X1 .. ., X ,, are independent and have distribution function F, thena, '(X,+ · · · +X.) + b ,, also has distribution function F.
=+28.10. Suppose that for alla, d',b, b' there exist a", b" (herea, a', a" are all positive)such that F(ax +b) . F(a'x + b') = F(a"x + b"). Show that F is stable.
=+28.11. Show that a stable law is infinitely divisible.
=+28.12. Show that the Poisson law, although infinitely divisible, is not stable.
=+28.13. Show that the normal and Cauchy laws are stable.
=+28.14. 28.101 Suppose that F has mean 0 and variance 1 and that the dependence of a", b" ona, a',b, b' is such that-
=+Show that F is the standard normal distribution.
=+28.15. (a) Let Y, be independent random variables having the Poisson distribution with mean cn"/|k|"+", where c > 0 and 0 (omit k =0 in the sum), and show that if c is properly chosen then the characteristic function of Z, converges to e Pr.
=+(b) Show for 0 < a ≤ 2 that e""""" is the characteristic function of a symmetric stable distribution; it is called the symmetric stable law of exponenta. The case a = 2 is the normal law, and a = 1 is the Cauchy law.
=+29.1. A real function f on R* is everywhere upper semicontinuous (see Problem
=+13.8) if for each x and € there is a 8 such that |x -y| < 8 implies that f(y)
=+(a) Use condition (iii) of Theorem 29.1, Fatou's lemma, and (21.9) to show that, if u, - u and f is bounded and lower semicontinuous, then(29.9)lim inf ffdu, ≥ ffdu.
=+(b) Show that, if (29.9) holds for all bounded, lower semicontinuous functionsf, then u, = p.
=+(c) Prove the analogous results for upper semicontinuous functions.
=+29.2. (a) Show for probability measures on the line that un X v1 = u Xv if and only if un = u and v => v.
=+(b) Suppose that X ,, and Y ,, are independent and that X and Y are indepen-dent. Show that, if X ,, = X and Y ,, = Y, then (X, Y) - (X, Y) and hence that X„ +Y, = X+Y.
=+(c) Show that part (b) fails without independence.
=+(d) If F. - F and G ,, - G, then F * G ,, = F . G. Prove this by part (b) and also by characteristic functions.
=+29.3. (a) Show that {u,) is tight if and only if for each € there is a compact set K such that w (K) > i -€ for all n.
=+(b) Show that {u ) is tight if and only if each of the k sequences of marginal distributions is tight on the line.
=+29.4. Assume of ( X, Y,) that X ,, = X and Y ,, =c. Show that ( X, Y ,, ) =(X,c). This is an example of Problem 29.2(b) where X ,, and Y ,, need not be assumed independent.
=+29.5. Prove analogues for R* of the corollaries to Theorem 26.3.
=+29.6. Suppose that f(X) and g(Y) are uncorrelated for all bounded continuous f and g. Show that X and Y are independent. Hint: Use characteristic func-tions.
=+29.7. 20.161 Suppose that the random vector X has a centered k-dimensional normal distribution whose covariance matrix has 1 as an eigenvalue of multi-plicity r and 0 as an eigenvalue of multiplicity k - r. Show that |X|' has the chi-squared distribution with r degrees of freedom.
=+29.8. 1 Multinomial sampling. Let p ....., p, be positive and add to 1, and let Z1, Z2 ,... be independent k-dimensional random vectors such that Z ,, has with probability p, a 1 in the ith component and 0's elsewhere. Then f ,, =(fm ,.... f.k) = Em _¡ Z ,, is the frequency count for a sample of
=+(a) Show that X ,, has mean values 0 and covariances 0;j = (8 ;; P; -
=+(b) Show that the chi squared statistic Ef_(f ., - np;)2 /np, has asymptotically the chi-squared distribution with k - 1 degrees of freedom.
=+29.9. 20.261 A theorem of Poincaré. (a) Suppose that X, = (X ,.,..., X ... ) is uniformly distributed over the surface of a sphere of radius in R". Fix t, and show that X ,.,..., X ,,, are in the limit independent, each with the standard normal distribution. Hint: If the components of Y ,, - (Y
=+(b) Suppose that the distribution of X ,, - (X ,.,,.. ., X ,.,, ) is spherically sym-metric in the sense that X ,, /IX ,, | is uniformly distributed over the unit sphere.Assume that [X ,, I /n =1, and show that X ,.,..., X ,., are asymptotically independent and normal.
=+29.10. Let X„, = (X .. ], . . ., X„ .. ), n = 1,2 ,..., be random vectors satisfying the mixing condition (27.19) with @ ,, = O(n ). Suppose that the sequence is stationary(the distribution of (X ,.,..., X ,, ,, ) is the same for all n), that E[ X ... ] =0, and that the Xmu are uniformly
=+29.11. 1 As in Example 27.6, let (Y) be a Markov chain with finite state space S = (1 ,..., s), say. Suppose the transition probabilities pu, are all positive and the initial probabilities p ., are the stationary ones. Let f ,,,, be the number of i for which 1 ≤i En and Y, = u. Show that the
=+has in the limit the centered normal distribution with covariances:00 Our -PuP. + E (PWP -P.P.) + E (PD)-P. P.).j-1 j-1
=+29.12. Assume that=b WOF22 is positive definite, invert it explicitly, and show that the corresponding two-di-mensional normal density is 11(29.10) f(x1, x2) = =2TD1/2 exp -2D(@22x2-2012x1x2+011x2) , where D = 011022 - 012.
=+29.13. Suppose that Z has the standard normal distribution in R'. Let u be the mixture with equal weights of the distributions of (Z, Z) and (Z, - Z), and let(X,Y) have distribution . Prove:
=+(a) Although each of X and Y is normal, they are not jointly normal.
=+(b) Although X and Y are uncorrelated, they are not independent.
=+30.1. From the central limit theorem under the assumption (30.5) get the full Lindeberg theorem by a truncation argument.
=+30.2. For a sample of size k ,, with replacement from a population of size n, the probability of no duplicates is I] == '(1 -j/n). Under the assumption kn/ Vn -0 in addition to (30.10), deduce the asymptotic normality of S, by a reduction to the independent case.
=+30.3. By adapting the proof of (21.24), show that the moment generating function of u in an arbitrary interval determines u.
=+30.4. 25.13 30.31 Suppose that the moment generating function of u ,, converges to that of u in some interval. Show that u ,, = ".
=+30.5. Let u be a probability measure on Rk for which /R+|x, l'u(dx) for i=1 ,..., k and r = 1,2 ,... . Consider the cross moments( ...... )= + ···? (dx)for nonnegative integers ri.
=+(a) Suppose for each i that(30.27)has a positive radius of convergence as a power series in 0. Show that a is determined by its moments in the sense that, if a probability measure v satisfies a(r1. ..., rk) = {x} ... x{v(dx) for all 1 ,.,, then coincides with u.
=+(b) Show that a k-dimensional normal distribution is determined by its mo-ments.
=+30.6. 1 Let u ,, and u be probability measures on R *. Suppose that for each i,(30.27) has a positive radius of convergence. Suppose that for all nonnegative integers r ...., rk. Show that H. p.
=+30.7. 30.5 1 Suppose that X and Y are bounded random variables and that X"and Y" are uncorrelated for m, n = 1, 2 ,... . Show that X and Y are indepen-dent.
=+30.8. 26.17 30.61 (a) In the notation (26.32), show for A # 0 that M[(cos Ax) ]=(+/2)2
=+(30.28)for even r and that the mean is 0 for odd r. It follows by the method of moments that cos Ax has a distribution in the sense of (25.18), and in fact of course the relative measure is 1
=+(30.29)p[ x : cos Ax ≤u] =1 -;arccos u,-1
=+(b) Suppose that A1, A2 ,. are linearly independent over the field of rationals in the sense that, if niÀ, + . . . +n",A ,, = 0 for integers nu, then n = . ="m =0. Show that M II(cos A,x) ` = HIM [(COSA,x)"]
=+(30.30)for nonnegative integers r1. ..., Fx.
=+(c) Let X1, X2 ,... be independent and have the distribution function on the right in (30.29). Show that(30.31)p| x: [ cos À x Su|= P[X] + ··· +X su].j=1(d) Show that WI
=+For a signal that is the sum of a large number of pure cosine signals with incommensurable frequencies, (30.32) describes the relative amount of time the signal is between u, and u2-
=+30.9. 6.161 From (30.16), deduce once more the Hardy-Ramanujan theorem (see(6.10)).
=+30.10. 1 (a) Prove that (if P ,, puts probability 1/n at 1 ,..., n)log log m - log log n(30.33)lim P. m:≥€ =0.ylog log n
=+(b) From (30.16) deduce that (see (2.35) for the notation)g (m) - log log m 1e-u2 /2 du.(30.34)D m:sx Vlog log m 12 TT 00
=+30.11. 1 Let G(m) be the number of prime factors in m with multiplicity counted.In the notation of Problem 5.19, G(m) = Epa (m).
=+(a) Show for k ≥ 1 that P,[m: @x (m) -8(m) >k] ≤ 1/pk+1; hence E ,, [a -8,]≤2/p2.(b) Show that E,[G - g] is bounded.
=+(c) Deduce from (30.16) that G(m) - log log n Sx\" 12- 1 -2 12 du .Pr m:Vlog log n 12 TT
=+(d) Prove for G the analogue of (30.34).
=+30.12. 1 Prove the Hardy-Ramanujan theorem in the form g(m)Dm:log log m - 1 2 € = 0.Prove this with G in place of g.
3. Consider the population consisting of all DVD players of a certain brand and model, and focus on whether a DVD player needs service while under warranty.a. Pose several probability questions based on selecting a sample of 100 such DVD players.b. What inferential statistics question might be
5. Many universities and colleges have instituted supplemental instruction (SI) programs, in which a student facilitator meets regularly with a small group of students enrolled in the course to promote discussion of course material and enhance subject mastery. Suppose that students in a large
6. The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border.A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss
7. A certain city divides naturally into ten district neighborhoods. Howmight a real estate appraiser select a sample of single-family homes that could be used as a basis for developing an equation to predict appraised value from characteristics such as age, size, number of bathrooms, distance to
8. The amount of ow through a solenoid valve in an automobile s pollution-control system is an important characteristic. An experiment was carried out to study how ow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor
9. In a famous experiment carried out in 1882, Michelson and Newcomb obtained 66 observations on the time it took for light to travel between two locations in Washington, D.C. A few of the measurements(coded in a certain manner) were 31, 23, 32, 36,2, 26, 27, and 31.a. Why are these measurements
10. Consider the IQ data given in Example 1.2.a. Construct a stem-and-leaf display of the data.What appears to be a representative IQ value?Do the observations appear to be highly concentrated about the representative value or rather spread out?b. Does the display appear to be reasonably symmetric
11. Every score in the following batch of exam scores is in the 60 s, 70 s, 80 s, or 90 s. A stem-and-leaf display with only the four stems 6, 7, 8, and 9 would not give a very detailed description of the distribution of scores. In such situations, it is desirable to use repeated stems. Here we
12. The accompanying speci c gravity values for various wood types used in construction appeared in the article Bolted Connection Design Values Based on European Yield Model (J. Struct. Engrg., 1993:2169—2186):.31 .35 .36 .36 .37 .38 .40 .40 .40.41 .41 .42 .42 .42 .42 .42 .43 .44.45 .46 .46 .47
13. The accompanying data set consists of observations on shower- ow rate (L/min) for a sample of n129 houses in Perth, Australia ( AnApplication of Bayes Methodology to the Analysis of Diary Records in a Water Use Study, J. Amer. Statist. Assoc., 1987:705—711):4.6 12.3 7.1 7.0 4.0 9.2 6.7 6.9
14. A Consumer Reports article on peanut butter (Sept.1990) reported the following scores for various brands:Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22 Crunchy 62 53 75 42 47 40 34 62 52 50 34 42 36 75 80 47 56 62 Construct a comparative stem-and-leaf display by listing stems
15. Temperature transducers of a certain type are shipped in batches of 50. A sample of 60 batches was selected, and the number of transducers in each batch not conforming to design speci cations was determined, resulting in the following data:2 1 2 4 0 1 3 2 0 5 3 3 1 3 2 4 7 0 2 3 0 4 2 1 3 1 1 3
16. In a study of author productivity ( Lotka s Test, Collection Manag., 1982: 111—118), a large number of authors were classi ed according to the number of articles they had published during a certain period.The results were presented in the accompanying frequency distribution:Number of papers 1
17. The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:Number of particles 0 1 2 3 4 5 6 7 Frequency 1 2 3 12 11 15 18 10 Number of particles 8 9 10 11 12 13 14

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