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modern mathematical statistics with applications

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

=+(a) Show that a trifling set is negligible.
=+(b) Show that the closure of a trifling set is also trifling.
=+(c) Find a bounded negligible set that is not trifling.
=+(d) Show that the closure of a negligible set may not be negligible.
=+(e) Show that finite unions of trifling sets are trifling but that this can fail for countable unions.
=+1.4. 1 For i = 0 ,..., r - 1, let A,(i) be the set of numbers in (0, 1] whose nonter-minating expansions in the base r do not contain the digit i.
=+(a) Show that A,(i) is trifling.
=+(b) Find a trifling set A such that every point in the unit interval can be represented in the form x + y with x and y in A.
=+(c) Let A,(i) ,..., i;) consist of the numbers in the unit interval in whose base-r expansions the digits i1 ,..., i, nowhere appear consecutively in that order.Show that it is trifling. What does this imply about the monkey that types at random?
=+1.5. 1 The Cantor set C can be defined as the closure of A3(1).
=+(a) Show that C is uncountable but trifling.
=+(b) From [0, 1] remove the open middle third (3, 3); from the remainder, a union of two closed intervals, remove the two open middle thirds ( ;, ? ) and(3, 8). Show that C is what remains when this process is continued ad infinitum.
=+c) Show that C is perfect [A15].
=+1.6. Put M(t) = (de15,(w) dw, and show by successive differentiations under the integral that(1.38)M(k)(0) = f's(w) do.Over each dyadic interval of rank n, s,(c) has a constant value of the form+1+1+ . .. + 1, and therefore M(t)=2-"Eexpt(+1+1+ . .. +1), where the sum extends over all 2" n-long
=+Use this and (1.38) to give new proofs of (1.16), (1.18), and (1.28). (This, the method of moment generating functions, will be investigated systematically in Section 9.)
=+1.7. 1 By an argument similar to that leading to (1.39) show that the Rademacher functions satisfy Sexp i Ear(w) do = []n 2| k=1 k=1 n= || cosa k .k=1
=+Take ak = 12-k, and from Ck=174(w)2-k = 2w - 1 deduce sin t = [] cost sin t(1.40)k=1 by letting n - oo inside the integral above. Derive Vieta's formula 2 12 1/2+12 12+12+2==2 22
=+1.8. A number wo is normal in the base 2 if and only if for each positive there exists an no(e, w) such that Ind(w)-
=+Theorem 1.2 concerns the entire dyadic expansion, whereas Theorem 1.1 concerns only the beginning segment. Point up the difference by showing that for € < ; the no(e, w) above cannot be the same for all @ in NV-in other words, n E" _, d,(w) converges to ; for all w in N, but not uniformly. But
=+1.9. 1.31 (a) Using the finite form of Theorem 1.3(ii), together with Problem
=+1.3(b), show that a trifling set is nowhere dense [A15].
=+(b) Put B = U, (r, - 2-n-2, r ,, + 2-" -2], where r1, r2 ,... is an enumeration of the rationals in (0, 1]. Show that (0, 1] - B is nowhere dense but not trifling or even negligible.
=+(c) Show that a compact negligible set is trifling.
=+1.10. 1 A set of the first category [A15] can be represented as a countable union of nowhere dense sets; this is a topological notion of smallness, just as negligibility is a metric notion of smallness. Neither condition implies the other:
=+(a) Show that the nonnegligible set N of normal numbers is of the first category by proving that Am = "-„,[w: In 's ,, (w)| < 2] is nowhere dense and NC Um Am.
=+(b) According to a famous theorem of Baire, a nonempty interval is not of the first category. Use this fact to prove that the negligible set Nc = (0, 1] - N is not of the first category.
=+1.11. Prove:(a) If x is rational, (1.33) has only finitely many irreducible solutions.
=+(b) Suppose that (q) ≥ 1 and (1.35) holds for infinitely many pairs p, q but only for finitely many relatively prime ones. Then x is rational.
=+(c) If » goes to infinity too rapidly, then A ,, is negligible (Theorem 1.6). But however rapidly o goes to infinity, A ,, is nonempty, even uncountable. Hint:Consider x = 24= 11/ 2ª(*) for integral @(k) increasing very rapidly to infinity.
=+Suppose that P is a probability measure on a field F, and that A, B E ៛and AC B. Since P(A) + P(B-A)=P(B), P is monotone:(2.5)P(A) ≤P(B)if ACB.
=+It follows further that P(B-A) = P(B) - P(A), and as a special case,(2.6)P(Ac) =1- P(A).
=+Other formulas familiar from the discrete theory are easily proved. For example,(2.7)
=+P(A) + P(B) =P(AUB)+P(ANB), the common value of the two sides being P(AUBC) +2P(AnB) + P(Ac)B). Subtraction gives
=+(2.8)P(AUB) = P(A) + P(B)-P(AnB).This is the case n = 2 of the general inclusion-exclusion formula:
=+(2.9) PUAK =EP(A) - EP(ANA,)k=1 i
=+n+1 nPU AX =P UAX |+P(AR+1) -P|U (ARNAn+1).k=1\k=1 k=1 Applying (2.9) to the first and third terms on the right gives (2.9) with n + 1 in place of n.
=+If B1 = A, and By = A , NAfn . . . nAL _, then the B, are disjoint and Ux -, AK = Ux_1 B ., so that P(U" , AK) ="_P(B). Since P(BA) ≤P(A]) by monotonicity, this establishes the finite subadditivity of P:P [ U AX|S CP(A).(2.10)k=1 k=1
=+2.1. Define x V y = max{x, y), and for a collection {x } define V x = sup_ x _;define x Ay = min(x, y) and A x = inf. x .. Prove that LAUB = IA V. IB, TANB= IA AIB, IA = 1 -14, and LAB =|I - Il, in the sense that there is equality at each point of 2. Show that A CB if and only if IA S IB
=+An (BUC)=(AnB) U (AnC). By similar arguments prove that AU(BNC) =(AUB)n(AUC), AAC C (AAB)U (BAC),(UA) = 1 4, n(nAn) = UA.n
=+2.2. Let A ,..., A ,, be arbitrary events, and put UR = U(A, n ... NA, ) and IK = n(A ,, U . . . UA,), where the union and intersection extend over all the k-tuples satisfying 1 si< ... < ik sn. Show that Uk = In-k+1.
=+2.3. (a) Suppose that 2€ F and that A, BE F implies A -B=ANB'EF.Show that $ is a field.
=+(b) Suppose that 2€ F and that } is closed under the formation of comple-ments and finite disjoint unions. Show that F need not be a field.
=+2.4. Let F1, F2, ... be classes of sets in a common space 2.
=+(a) Suppose that 7 ,, are fields satisfying $, C $ ,., . Show that U"-19 ,, is a field.
=+(b) Suppose that $ ,, are o-fields satisfying 9, 9n+1. Show by example that Un-19, need not be a o-field.
=+2.5. The field f(. ) generated by a class & in 0 is defined as the intersection of all fields in 2 containing .
=+(a) Show that f(.) is indeed a field, that cf(), and that f() is minimal in the sense that if 9 is a field and CO, then f(A) c.9.
=+(b) Show that for nonemptya, f() is the class of sets of the form Un ;, A ., where for each i and j either A, E & or AGE, and where the m sets n ,, A .,, 1 sism, are disjoint. The sets in f( ) can thus be explicitly presented, which is not in general true of the sets in o( ! ).
=+. 1 (a) Show that if & consists of the singletons, then f(.) is the field in Example 2.3.
=+(b) Show that f() Co(), that f() =() if is finite, and that a(f(e) = 0 (x).
=+(c) Show that if & is countable, then f(. ) is countable.
=+(d) Show for fields F1 and F2 that f(9, U 2) consists of the finite disjoint unions of sets A, NA2 with A, E F. Extend.
=+2.7. 2.5 Let H be a a set lying outside $, where " is a field [or o-field]. Show that the field [or o-field] generated by FU [H] consists of sets of the form
=+2.8. Suppose for each A in & that A' is a countable union of elements of &. The class of intervals in (0, 1] has this property. Show that a() coincides with the smallest class over & that is closed under the formation of countable unions and intersections.
=+2.9. Show that, if BEo(), then there exists a countable subclass & of & such that BEO(AB).
=+2.10. (a) Show that if (0) contains every subset of 0, then for each pair w and w'of distinct points in n there is in & an A such that I,(w) # 14(w).
=+(b) Show that the reverse implication holds if 22 is countable.
=+(c) Show by example that the reverse implication need not hold for uncount-able 22.
=+2.11. A o-field is countably generated, or separable, if it is generated by some countable class of sets.
=+(a) Show that the o-field $ of Borel sets is countably generated.
=+(b) Show that the o-field of Example 2.4 is countably generated if and only if S is countable.
=+(c) Suppose that F1 and 2 are o-fields, FC F2, and $2 is countably generated. Show by example that 7, may not be countably generated.
=+2.12. Show that a o-field cannot be countably infinite-its cardinality must be finite or else at least that of the continuum. Show by example that a field can be countably infinite.
=+2.13. (a) Let F be the field consisting of the finite and the cofinite sets in an infinite 2, and define P on § by taking P( A) to be 0 or 1 as A is finite or cofinite.(Note that P is not well defined if 2 is finite.) Show that P is finitely additive.
=+(b) Show that this P is not countably additive if 22 is countably infinite.
=+(c) Show that this P is countably additive if 02 is uncountable.
=+(d) Now let 9 be the o-field consisting of the countable and the cocountable sets in an uncountable 2, and define P on } by taking P( A) to be 0 or 1 as A is countable or cocountable. (Note that P is not well defined if 22 is countable.)Show that P is countably additive.
=+2.14. In (0, 1] let } be the class of sets that either (i) are of the first category [A15] or(ii) have complement of the first category. Show that F is a o-field. For A in
=+F, take P(A) to be 0 in case (i) and 1 in case (ii). Show that P is countably additive.
=+2.15. On the field $o in (0, 1] define P( A) to be 1 or 0 according as there does or does not exist some positive €4 (depending on A) such that A contains the interval ( ¿, ¿ + €4]. Show that P is finitely but not countably additive. No such example is possible for the field Go in S®
=+2.16. (a) Suppose that P is a probability measure on a field §. Suppose that A, € ៛for t > 0, that A, CA, for s < t, and that A = U, > A, E F. Extend Theorem
=+2.1(i) by showing that P(A,){ P( A) as t -> co. Show that A necessarily lies in if it is a o-field.(b) Extend Theorem 2.1(ii) in the same way.
=+2.17. Suppose that P is a probability measure on a field ", that A1, A2 ,..., and A = U ,, A, lie in F, and that the A ,, are nearly disjoint in the sense that P(A„OA„) =0 for m + n. Show that P(A) =C ,, P(A ,, ).2.18. Stochastic arithmetic. Define a set function P ,, on the class of all
=+P ., ( A) = = #[m: 1 ≤m ≤n, m€A];among the first n integers, the proportion that lie in A is just P.( A). Then P, is a discrete probability measure. The set A has density(2.35)D(A) = lim P„(A), provided this limit exists. Let 9 be the class of sets having density.
=+(a) Show that D is finitely but not countably additive on 9.
=+(b) Show that 9 contains the empty set and $2 and is closed under the formation of complements, proper differences, and finite disjoint unions, but is not closed under the formation of countable disjoint unions or of finite unions that are not disjoint.
=+(c) Let . consist of the periodic sets Ma = [ka: k = 1, 2, ... ]. Observe that P1(M)=||||→t=D(M).-10(2.36)
=+Show that the field f(. ) generated by .A (see Problem 2.5) is contained in 9.
=+Show that D is completely determined on f(. ) by the value it gives for each a to the event that m is divisible by a.
=+d) Assume that Ep- diverges (sum over all primes; see Problem 5.20(e)) and prove that D, although finitely additive, is not countably additive on the field f(M).
=+(e) Euler's function (n) is the number of positive integers less than n and relatively prime to it. Let p1 ,..., p, be the distinct prime factors of n; from the inclusion-exclusion formula for the events [m: p; Im], (2.36), and the fact that the p, divide n, deduce 4(n)- п(1-1).(2.37)n pln
=+(f) Show for 0 ≤ x ≤ 1 that D( A) =x for some A.
=+(g) Show that D is translation invariant: If B = [m + 1: m € A], then B has a density if and only if A does, in which case D( A) = D(B).
=+2.19. A probability measure space (2, 7, P) is nonatomic if P(A) > 0 implies that there exists a B such that B CA and 0 < P(B) < P(A) (A and B in }, of course).
=+(a) Assuming the existence of Lebesgue measure À on Ø, prove that it is nonatomic.
=+(b) Show in the nonatomic case that P(A) > 0 and € > 0 imply that there exists a B such that B CA and 0< P(B)
=+c) Show in the nonatomic case that 0 ≤ x ≤ P( A) implies that there exists a B such that B CA and P(B) =x. Hint: Inductively define classes & ,, numbers h, and sets H, by X = (Ø) = (H), * = [H: H CA - Uk
=+d) Show in the nonatomic case that, if p1, P2, . .. are nonnegative and add to 1, then A can be decomposed into sets B1, B2, ... such that P(B ,, ) = p ,, P(A).
=+2.20. Generalize the construction of product measure: For n = 1, 2 ,..., let S ,, be a finite space with given probabilities P ..., u E S ,,. Let S, x S2 X . . . be the space of sequences (2.15), where now z,(@) ES ,. Define P on the class of cylinders, appropriately defined, by using the product
=+2.21. (a) Suppose that = (A), A2 ,... ) is a countable partition of n. Show (see
=+(2.27) that = *=* coincides with (). This is a case where ()can be constructed "from the inside."(b) Show that the set of normal numbers lies in .(c) Show that ** = " if and only if W is a g-field. Show that 1 ; is strictly smaller than $, for all n.
=+2.22. Extend (2.27) to infinite ordinals a by defining . = (U 3 < a 3) *. Show that, if 2 is the first uncountable ordinal, then U. < na =o(A). Show that, if the cardinality of & does not exceed that of the continuum, then the same is true of a(). Thus S has the power of the continuum.
=+2.23. 1 Extend (2.29) to ordinals & < 2 as follows. Replace the right side of (2.28)by Un- ((A2, -, VAS ,, ). Suppose that $g is defined for B
=+define(2.38) Фа(А1, А2 ,... )= {(B.(1)( Am) Amı2) ... ),$$„(2)( Am2,› Am2" ... ) ,... ).
=+Prove by transfinite induction that (2.38) is in @ if the A ,, are, that every element of A has the form (2.38) for sets A ,, in ,, and that (2.31) holds with a in place of n. Define ( (w) = $ (I .,, I .,,... ), and show that B =[w.w 4 (w)] lies in - for a < Q. Show that is strictly smaller than
=+Let P be a probability measure on a field $7. The construction following extends P to a class that in general is much larger than o(5%) but nonethe-less does not in general contain all the subsets of f.For each subset A of 22, define its outer measure by
=+(3.1)P*(A) = inf _P(A.), where the infimum extends over all finite and infinite sequences A1, A2 ,...of So-sets satisfying AC U ,, A ,,. If the A ,, form an efficient covering of A, in the sense that they do not overlap one another very much or extend much
=+beyond A, then E ,, P(A ,, ) should be a good outer approximation to the measure of A if A is indeed to have a measure assigned it at all. Thus (3.1)represents a first attempt to assign a measure to A.Because of the rule P(Ac)=1 - P(A) for complements (see (2.6)), it is
=+natural in approximating A from the inside to approximate the complement Ac from the outside instead and then subtract from 1:(3.2)P.(A)=1-P*(Ac).
=+This, the inner measure of A, is a second candidate for the measure of A.' A plausible procedure is to assign measure to those A for which (3.1) and (3.2)'An idea which seems reasonable at first is to define P. (4) as the supremum of the sums" ,, P(A ) for disjoint sequences of $7-sets in A. This

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