Probability And Measure Wiley Series In Probability And Mathematical Statistics 3rd Edition Patrick Billingsley - Solutions

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76.a. Use the general formula for the variance of a linear combination to write an expression for V(aX Y). Then let a sY /sX, and show that r 1. [Hint: Variance is always 0, and Cov(X, Y) sX # sY # r.]b. By considering V(aX Y), conclude that r 1.c. Use the fact that V(W) 0 only if Wis
74. Refer to Exercise 27, and suppose that the Xi s are independent with each one having a normal distribution.What is the probability that the total volume shipped is at most 100,000 ft3?
73. The amount of soft drink that Ann consumes on any given day is independent of consumption on any other day and is normally distributed with m13 oz and s 2. If she currently has two six-packs of 16-oz bottles, what is the probability that she still has some soft drink left at the end of 2
70. We have seen that if E(X1)E(X2). . .E(Xn)m, then E(X1. . .Xn)nm. In some applications, g1Xi Yi Zi 2 the number of Xi s under consideration is not a xed number n but instead is an rv N. For example, let Nthe number of components that are brought into a repair shop on a particular
68. Suppose that for a certain individual, calorie intake at breakfast is a random variable with expected value 500 and standard deviation 50, calorie intake at lunch is random with expected value 900 and standard deviation 100, and calorie intake at dinner is a random variable with expected value
67. In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information about the total cost distribution by adding together characteristics of the individual component
65.a. Use Exercise 47 to approximate , and compare the result with the answer given by software, .03237.b. Use the formula given at the bottom of Table A.7,, to approximate, and compare with part (a).
64. Let Z1, Z2, . . . , Z10 be independent standard normal.Use these to constructa. A random variableb. A t4 random variablec. An F4,6 random variabled. A Cauchy random variablee. An exponential random variable with mean 2f. An exponential random variable with mean 1 g. A gamma random variable with
63. Show that if X is gamma distributed and c(0) is a constant, then cX is gamma distributed. In particular, if X is chi-squared distributed, then cX is gamma distributed.
62.a. Use Table A.8 to nd t.25,10.b. Use (a) to nd the median of F1,10.c. Verify the answer to part (b) using software (e.g., TI-89 calculator or MINITAB).
61.a. Use Table A.9 to nd F.1,2,4.b. Verify the answer to part (a) using the pdf.c. Verify the answer to part (a) using software (e.g., TI-89 calculator or MINITAB).
60. Derive the F pdf by applying the method used to derive the t pdf.
59. Show that .
58. Is it true that ?Explain.
57. Let X have an F distribution with n1 numerator df and n2 denominator df.a. Determine the mean value of X.b. Determine the variance of X.
56. Show that the ratio of two independent standard normal random variables has the t1 distribution. Apply the method used to derive the t pdf in this section.Hint: Split the domain of the denominator into positive and negative parts.1x 1/22/ 31x1x2 4 S1 X2 x2n 3n1 X1 x2n 1, X3 x2n 3E1S2
55. Show directly from the pdf that the mean of a t1(Cauchy) random variable does not exist.
54. Show that the t pdf approaches the standard normal pdf for large df values. Hint: Use (1 a/x)x S ea and as xSq.
53.a. Use Table A.8 to nd t.005,10.b. Use Table A.9 to nd F.01,1,10 and relate this to the value you obtained in part (a).c. Verify the answer to part (b) using software (e.g., TI-89 calculator or MINITAB).
52.a. Use Table A.8 to nd t.102,1.b. Verify the answer to part (a) by integrating the pdf.c. Verify the answer to part (a) using software (e.g., TI-89 calculator or MINITAB).
51. Use moment generating functions to show that if X3X1X2,with ,n3 n1, and X1 and X2 are independent, then .
50. Knowing that for a normal random sample,a. Show that E(S2) s2.b. Show that V(S2) 2s4/(n 1). What happens to this variance as n gets large?c. Apply Equation (6.12) to show that Then show that if n 2. Is it true that E(S) s for normal data?
49. Show that the pdf has a maximum at n 2 if n2.
48. Apply the Law of Large Numbers to show that approaches 1 as n becomes large.
47. Why should be approximately normal for large n? What theorem applies here, and why?
46.a. Use Table A.7 to nd .b. Verify the answer to (a) by integrating the pdf.c. Verify the answer to (a) by using software (e.g., TI-89 calculator or MINITAB).
45. The proof of the Central Limit Theorem requires calculating the moment generating function for the standardized mean from a random sample of any distribution, and showing that it approaches the moment generating function of the standard normal distribution.Here we look at a particular case of
44. Let X and Y be independent gamma random variables, both with the same scale parameterb. The value of the other parameter is a1 for X and a2 for Y.Use moment generating functions to show that X Y is also gamma distributed with scale parameter b, X Y15 P1X Y152 P12 X Y 12.and with the
42. For males the expected pulse rate is 70 per second and the standard deviation is 10 per second. For women the expected pulse rate is 77 per second and the standard deviation is 12 per second. Let the sample average pulse rate for a random sample of 40 men and let the sample average pulse
41. A professor has three errands to take care of in the Administration Building. Let Xi the time that it takes for the ith errand (i 1, 2, 3), and let X4 the total time in minutes that she spends walking to and from the building and between each errand. Suppose the Xi s are independent,
=+31.1. Extend Examples 31.1 and 31.3: Let Por ..., p, ; be nonnegative numbers adding to 1, where r ≥ 2; suppose there is no i such that p, = 1. Let X1,
=+X2 ,... be independent, identically distributed random variables such that P[X] = i]=p ;, 0≤i
=+31.3. What is the analogue of (31.17)?
=+31.2. 1 In Problem 31.1 take r = 3 and Po =P2= ;, p1 = 0. The corresponding F is called the Cantor function. The complement in [0, 1] of the Cantor set (see Problems 1.5 and 3.16) consists of the middle third ( 3, 3), the middle thirds
=+(, ) and (#. #), and so on. Show that F is { on the first of these intervals, ;on the second, 2 on the third, and so on. Show by direct argument that F' = 0 except on a set of Lebesgue measure 0.
=+31.3. A real function f of a real variable is a Lebesgue function if [x: f(x) ≤a] is a Lebesgue set for each a.
=+(a) Show that, iff, is a Borel function and f2 is a Lebesgue function, then the compositionf, f2 is a Lebesgue function.
=+(b) Show that there exists a Lebesgue functionf, and a Lebesgue (even Borel, even continuous) function f2 such that f1 f2 is not a Lebesgue function. Hint:Use Example 31.4.
=+31.4. 1 An arbitrary function f on (0, 1] can be represented as a composition of a Lebesgue functionf, and a Borel function f2. For x in (0, 1], let d (x) be the nth digit in its nonterminating dyadic expansion, and define f2(x) =
=+... , 2d,(x)/3". Show that f2 is increasing and that f2(0, 1] is contained in the Cantor set. Take f (x) to be f( f2'(x)) if x €f2(0, 1] and 0 if x € (0,1]-f,(0,1].Now show that f=fif2.
=+31.5. Let r1, r2 ,... be an enumeration of the rationals in (0, 1) and put F(x) =Ex:r. < , 2 -*. Define pp by (14.5) and prove that it is continuous and singular.
=+31.6. Suppose that u and F are related by (31.18). If F is not absolutely continuous, then p( A) > 0 for some set A of Lebesgue measure 0. It is an interesting fact, however, that almost all translates of A must have u-measure 0. From Fubini's theorem and the fact that À is invariant under
=+31.7. 17.4 31.61 Show that F is absolutely continuous if and only if for each Borel set A. u( A + x) is continuous in x.
=+31.8. Let F. (x) = lim ; - ( inf (F(u) - F(u))/(v -u), where the infimum extends over u and u such that u
=+31.9. Lebesgue's density theorem. A point x is a density point of a Borel set A if A(u, u]nA)/(u-u) -> 1 as uf x and v 1 x. From Theorems 31.2 and 31.4 deduce that almost all points of A are density points. Similarly, A(u, u]n A)/(v - u) - 0 almost everywhere on Ac.
=+31.10. Let f: [a, b] - R* be an arc; f(t) =(f;(t) ,..., f,(r)). Show that the arc is rectifiable if and only if eachf, is of bounded variation over [a, b].
=+31.11. 1 Suppose that F is continuous and nondecreasing and that F(0) = 0, F(1)=1. Then f(x)=(x, F(x)) defines an arc f: [0,1] -R2. It is easy to see by monotonicity that the arc is rectifiable and that, in fact, its length satisfies
=+L(f)≤2. It is also easy, given €, to produce functions F for which L(f)> 2-€.Show by the arguments in the proof of Theorem 31.4 that L(f) = 2 if F is singular.
=+31.12. Suppose that the characteristic function of F satisfies lim sup, __ lo(t)| = 1.Show that F is singular. Compare the lattice case (Problem 26.1). Hint: Use the Lebesgue decomposition and the Riemann-Lebesgue theorem.
=+31.13. Suppose that X1, X2 ,... are independent and assume the values +1 with probability ; each, and let X = 2- 1X ,, /2". Show that X is uniformly dis-
=+tributed over [-1, +1]. Calculate the characteristic functions of X and X, and deduce (1.40). Conversely, establish (1.40) by trigonometry and conclude that X is uniformly distributed over [-1, + 1].
=+31.14. (a) Suppose that X1, X2 ,... are independent and assume the values 0 and 1 with probability ; each. Let F and G be the distribution functions of
=+2n=1X2m - 1/22m=1 and L"_ X2,/22". Show that F and G are singular but that F * G is absolutely continuous.
=+(b) Show that the convolution of an absolutely continuous distribution func-tion with an arbitrary distribution function is absolutely continuous.
=+31.15. 31.21 Show that the Cantor function is the distribution function of Zu - 1X„/3", where the X ,, are independent and assume the values 0 and 2 with probability ; each. Express its characteristic function as an infinite product.
=+31.16. Show for the F of Example 31.1 that ,D(1) - and DF(0) -0 if Po < 2. From (31.17) deduce that - D(x) => and D'(x) =0 for all dyadic rationals x. Analyze the case po > 2 and sketch the graph.
=+31.17. 6.141 Let F be as in Example 31.1, and let u be the corresponding probability measure on the unit interval. Let d.(x) be the nth digit in the nonterminating binary expansion of x, and let s,(x) = Ex _, d,(x). If I (x) is
=+the dyadic interval of order n containing x, then s ,. (x(31.33)- - log u (1,(x) = - |1 -log Po log Pi-
=+(a) Show that (31.33) converges on a set of u-measure 1 to the entropy h = - po log po - p, log p1. From the fact that this entropy is less than log2 if Po +5, deduce in this case that on a set of u-measure 1, F does not have a finite derivative.
=+(b) Show that (31.33) converges to - ¿ log po - ¿ log p, on a set of Lebesgue measure 1. If po = = this limit exceeds log2 (arithmetic versus geometric means), and so u(/„(x))/2-" -> 0 except on a set of Lebesgue measure 0. This does not prove that F'(x) exists almost everywhere, but it does
=+(c) Show that, if (31.33) converges to /, then if @ > 1/log2, 00 (31.34)lim 0 if a
=+If (31.34) holds, then (roughly) F satisfies a Lipschitz condition™ of (exact)order //log 2. Thus F satisfies a Lipschitz condition of order h/log 2 on a set of u-measure 1 and a Lipschitz condition of order ( - ¿ log po - ¿ log p,)/log 2 on a set of Lebesgue measure 1.
=+31.18. van der Waerden's continuous, nowhere differentiable function is f(x)=Ex-og (x), where ag(x) is the distance from x to the nearest integer and a.(x) = 2-ag(2"x). Show by the Weierstrass M-test that f is continuous. Use
=+(31.8) and the ideas in Example 31.1 to show that f is nowhere differentiable.'A Lipschitz condition of order @ holds at x if F(x + h) - F(x) - O(|h|") as h -> 0; for @ > 1 this implies F'(x) =0, and for 0
=+31.19. Show (see (31.31)) that (apart from addition of constants) a function can have only one representation F, + F2 with F, absolutely continuous and F2 singu-lar.
=+31.20. Show that the F, in the Lebesgue decomposition can be further split into Fa + F ., where F _, is continuous and singular and Fa increases only in jumps in the sense that the corresponding measure is discrete. The complete decom-position is then F = Fac + Fc3 + Fa-
=+31.21. (a) Suppose that x ,-< x2 < .. . and , IF(x,) =c. Show that, if F assumes the value 0 in each interval (x ,,, x ,,, , ), then it is of unbounded variation.
=+(b) Define F over [0, 1] by F(0) = 0 and F(x) =xª sin x"1 for x > 0. For which values of « is F of bounded variation?
=+31.22. 14.41 If f is nonnegative and Lebesgue integrable, then by Theorem 31.3 and (31.8), except for x in a set of Lebesgue measure 0, 1(31.35)['f(t)di-f(x)v -u
=+if u ≤x ≤u, u x. There is an analogue in which Lebesgue measure is replaced by a general probability measure u: If f is nonnegative and integrable with respect to u, then as h 10, 1(31.36)f(t)p(dt) -f(x)«(x-h,x+h) (x-h.x+k]
=+on a set of u-measure 1. Let F be the distribution function corresponding to", and put (u) = inf[ x: u ≤ F(x)] for 0
=+32.1. There are two ways to show that the convergence in (32.1) must be absolute:Use the Jordan decomposition. Use the fact that a series converges absolutely if it has the same sum no matter what order the terms are taken in.
=+32.2. If A+UA" is a Hahn decomposition of p, there may be other ones AUAF.Construct an example of this. Show that there is uniqueness to the extent that"(AA+)="(A"&A;) =0.
=+32.3. Show that absolute continuity does not imply the €-8 condition (32.4) if v is infinite. Hint: Let } consist of all subsets of the space of integers, let v be counting measure, and let u have mass n=2 at n. Note that u is finite and v is o-finite.
=+32.4. Show that the Radon-Nikodym theorem fails if u is not o-finite, even if v is finite. Hint: Let F consist of the countable and the cocountable sets in an
=+uncountable 22, let u be counting measure, and let v( A) be 0 or 1 as A is countable or cocountable.
=+32.5. Let u be the restriction of planar Lebesgue measure À, to the o-field F={A XR1: A € R1} of vertical strips. Define v on F by v( AX R') = A2(A X (0, 1)). Show that v is absolutely continuous with respect to u but has no density. Why does this not contradict the Radon-Nikodym theorem?
=+32.6. Let u, D, and p be o-finite measures on (2, 5 ). Assume the Radon-Nikodym derivatives here are everywhere nonnegative and finite.(a) Show that >
=+(c) Suppose that u 0] du/dp
=+32.7. Show that there is a Lebesgue decomposition (32.8) in the o-finite as well as the finite case. Prove that it is unique.
=+32.8. The Radon-Nikodym theorem holds if u is o-finite, even if v is not. Assume at first that u is finite (and > < p).
=+(a) Let $ be the class of (-sets) B such that u(E) = 0 or v(E) = œ for each E c B. Show that $ contains a set Bo of maximal u-measure.
=+(b) Let € be the class of sets in 22g = BS that are countable unions of sets of finite v-measure. Show that 6 contains a set Co of maximal u-measure. Let Do=225 - Co.
=+(c) Deduce from the maximality of Bo and Co that u( Dg) = v(D) =0.
=+(d) Let vo( A) =v(Ann,). Using the Radon-Nikodym theorem for the pair H, Vo, prove it for u, v.
=+e) Now show that the theorem holds if p is merely o-finite.
=+(f) Show that if the density can be taken everywhere finite, then v is o-finite.
=+32.9. Let u and v be finite measures on (2, 9 ), and suppose that " is a o-field contained in $. Then the restrictions w" and " of u and v to Fº are measures on (n, º). Let Vac, "g, Vic, Vy be, respectively, the absolutely contin-uous and singular parts of v and vº with respect to u and º.
=+. Suppose that u, v, v ,, are finite measures on (2, 9 ) and that v( A) = E ,, v (A)for all A. Let v.( A) = f.f ., du + v"( A) and v( A) = [Afdu + v'(A) be the decompositions (32.8); here v' and », are singular with respect to u. Show that
=+f =E,f, except on a set of u-measure 0 and that v'(A) = E ,, v'(4) for all A.Show that > < u if and only if v, < u for all n.
=+32.11. 32.2 | Absolute continuity of a set function ø with respect to a measure u is defined just as if p were itself a measure: p( A) = 0 must imply that "( A) = 0.Show that, if this holds and u is o-finite, then "( A) = [ fdu for some
=+integrablef. Show that A+=[w: f(w) > 0] and A =[@: f(w) < 0] give a Hahn decomposition for . Show that the three variations satisfy @ +(A) =Jaf' du, "(A) = [Af"du, and |opKA) = JAlfl du. Hint: To constructf, start with (32.2).
=+32.12. 1 A signed measure p is a set function that satisfies (32.1) if A1, A2 ,... are disjoint and may assume one of the values + oo and -o but not both. Extend the Hahn and Jordan decompositions to signed measures.
=+32.13. 31.221 Suppose that u and v are a probability measure and a o-finite measure on the line and that v < u. Show that the Radon-Nikodym derivative f satisfies v(x -h, x +h]lim*= f(x)h-cp(x-h, x + h]on a set of u-measure 1.
=+32.14. Find on the unit interval uncountably many probability measures up, 0
=+32.15. Let $, be the field consisting of the finite and the cofinite sets in an uncountable 22. Define , on 9, by taking ( A) to be the number of points in
=+A if A is finite, and the negative of the number of points in A' if A is cofinite.Show that (32.1) holds (this is not true if 22 is countable). Show that there are no negative sets for o (except the empty set), that there is no Hahn decomposi-tion, and that ? does not have bounded range.
=+33.1. 20.27 1 Borel's paradox. Suppose that a random point on the sphere is specified by longitude 0 and latitude $, but restrict O by 0 ≤ 0 < T, so that 0 specifies the complete meridian circle (not semicircle) containing the point, and compensate by letting @ range over (-Tr, 7].
=+(a) Show that for given @ the conditional distribution of $ has density Licos | over ( -w, + w]. If the point lies on, say, the meridian circle through Greenwich, it is therefore not uniformly distributed over that great circle.
=+(b) Show that for given $ the conditional distribution of @ is uniform over(0, T). If the point lies on the equator ($ is 0 or Tr), it is therefore uniformly distributed over that great circle.

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Probability And Measure Wiley Series In Probability And Mathematical Statistics 3rd Edition Patrick Billingsley - Solutions

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