New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
elementary probability for applications

A Course In Probability Theory 3rd Edition Kai Lai Chung - Solutions

1.*2. Let {Xn} be independent and identically distributed with mean 0 and variance I. Then we have for every x:[HINT: Let:J'>{ max S j 2: x} :s 2g7>{S n 2: x -J2,;}.lS;.jS;.n Ak = {max Sj < x; Sk > x}lS;.j
1.1. Theorem S.3.l has the following "one-sided" analogue. Under the same hypotheses, we have [This is due to A. W. Marshall.]
1.14. Let {bn} be as in Theorem 5.2.3. and put Xn = 2bll for n ~ 1. Then there exists {all} for which (6) holds, but condition (i) does not hold Thus condition (7) cannot be omitted.
1.13. Let {Xn} be a sequence of identically distributed strictly positive random variables. For any cp such that cp(n )/n ---+ 0 as n ---+ 00, show that JO{SII > cp(n) i.o.} = 1, and so Sn ---+ 00 a.e. [HINT: Let Nn denote the number of k ~ 11 such that Xk ~ cp(n)ln. Use Chebyshev'S inequality to
1.12. Theorem 5.2.2 may be slightly generalized as follows. Let {Xn} be pairwise independent with a common d.f. F such that(i)1 x dF(x) = 0(1), Ixl:511(ii) n 1 dF(x) = 0(1);Ixl>n then SI1 111 ---+ 0 in pr.
1.11. Derive the following form of the weak law of large numbers from Theorem 5.2.3. Let {bll }be as in Theorem 5.2.3 and put Xn = 2bn for n ::::l.Then there exists {an} for which (6) holds but condition (i) does not.
1.*10. I,et {Xn, 1 < n < oo} be arbitrary r v's and for each n let mn be a median of Xn. Prove that if X/J ---+ Xoo in pro and moo is unique, then mn ---+moo. Furthermore, if there exists any sequence of real numbers {en} such that Xn -en ---+ 0 in pr., then Xn -mil ---+ 0 in pr.
1.9. A median of the r.v. X is any number a such that?{ X < ex} > 1 9{ X > ex} > 1 2' 2 Show that such a number always exists but need not be unique.
1.8. They also imply that[HINT: Use the first part of Exercise 7 and divide the interval of integration hJ< Ixl ~ hn into parts of the form )..k < Ixl ~ )..k+l with).. > 1.]
1.7. Conditions (i) and (ii) in Theorem 5.2.3 imply that for any 0 > 0, and that all = o (.j1ibll ).
1.* 6. Show on the contrary that a weak law of large numbers does hold for bn = n log 11 and find the corresponding an. [HINT: Apply Theorem 5.2.3.]
1.*5. Let .:'7>(Xl = 2n) = 1j2n, n :::: 1; and let {Xn, n :::: I} be independent and identically distributed. Show that the weak law of large numbers does not hold for bll= 11; namely, with this choice of bn no sequence {an} exists for which (6) is true. [This is the St. Petersburg paradox, in
1.* 4. For any 0 > 0, we have uniformly in p: 0 < p < l.
1.3. For any sequence {Xn}:Sn 0 . XII 0 . -~ In pr. =} -~ In pro n n More generally, thh is true if n is replaced by bn, where bn+l/bll~ 1.
1.2. Even for a sequence of independent r.v.'s {Xn}, X O· ~~ O· n ~ III pr. -r-r -~ III pro n[HINT: Let Xn take the values 211 and 0 with probabilities n-I and 1 -11-1 .]
1.1. For any sequence of r.v.'s {Xn}, and any p 2: 1:Sn Xn ~ 0 a.e. =} -~ 0 a.e., n X n ~ 0 in L P :::::} Sn ~ 0 in L P •n The second result is false for p < 1.
1.*10. Is the sum of two normal numbers, modulo 1, normal? Is the product?[HINT: Consider the differences between a fixed abnormal number and all normal numbers: this is a set of probability one.]
1.9. Prove that the set of real numbers in [0, 1] whose decimal expansions do not contain the digit 2 is of measure zero. Deduce from this the existence of two sets A and B both of measure zero such that every real number is representable as a sum a + b with a E A, b E B.
1.*8. Let X be an arbitrary r.v. with an absolutely continuous distribution.Prove that with probability one the fractional part of X is a normal number.[HINT: Let N be the set of normal numbers and consider g'){X -[X] EN}.]
1.7. Let a be completely normal. Show that by looking at the expansion of a in some scale we can rediscover the complete works of Shakespeare from end to end without a single misprint or interruption. [This is Borel's paradox.]
1.* 6. The above definition may be further strengthened if we consider diffe rent scales of expansion. A real number in [0, 1] is said to be complezely nonnal iff the relative frequency of each block of length r in the scale s tends to the limit I/sr for every sand r. Prove that almost every number
1.5. We may strengthen the definition of a normal number by considering blocks of digits. Let r ::::. 1, and consider the successive overlapping blocks of r consecutive digits in a decimal; there are n -r + 1 such blocks in the first 11 places. Let V(II)(W) denote the number of such blocks that are
1.* 4. If {XII} are independent r. v.' s such that the fourth moments (r (X~ )have a common bound, then (1) is true a.e. [This is Cantelli's strong law of large numbers. Without using Theorem 5.1.2 we may operate with f (S~ I n4 )as we did with /(S~/n2). Note that the full strength of independence
1.3. Theorem 5.1.2 remains true if the hypothesis of bounded second mo ments is weakened to: 0'2(Xn) = 0(n8 ) where° ::s e < k. Various combina tions of Exercises 2 and 3 are possible.
1.*2. Theorem 5.1.2 may be sharpened as follows: under the same hypo theses we have SII /na -+° a.e. for any a > ~.
1.1. FOI any sequence of I. v.' s {XII}, if (r(X~) ----+ 0, then (l) is tlUe in pI.but not necessarily a.e.
1.*10. Suppose the distributions of {Xn, 1
1.9. If {Xn} is uniformly integrable, then the sequence is uniformly integrable.
1.*8. If sUPn {(IXn IP) < 00 for some p > 1, then {Xn} is uniformly inte grable.
1.7. If {X n} is dominated by some Y in L 1 or if it is identically distributed with finite mean, then it is uniformly integrable.4.5 UNIFORM INTEGRABILITY; CONVERGENCE OF MOMENTS I 105
1.6. If {Xt} and {Yt} are uniformly integrable, then so is {Xt+Yd and{Xt+Yt}.
1.5. Find the moments of the normal dJ. and the positive normal dJ.+ below:if x > o·if x < O.Show that in either case the moments satisfy Carleman's condition.
1.*4. Exercise 3 may be reduced to Exercise 10 of Sec. 4.1 as follows. Let F 11' 1 < n < 00, be dJ.' s such that F 11 ~ F. Let 8 be uniformly distributed on[0, 1] and put Xn = F-;;l (8), 1 ::s n < 00, where F-;;l (y) = sup{x: F n(x) < y}(cL Exercise 4 of Sec. 3.1). Then Xn has dJ. F n and Xn ----+
1.3. If XII ---+ X in dist., and fEe, then f (Xn) ---+ f (X) in dist.
1.2. If {X II} is dominated by some Y in LP, and converges in dist. to X,
1.1. If supn IXn I E LP and Xn ---+ X a.e., then X E LP and Xn ---+ X in LP.
1.*12. Let Fn and F be dJ.'s such that Fn ~ F. Define Gn(e) and G(e)as in Exercise 4 of Sec. 3.1. Then Gn (e) ---+ G(e) in a.e. [HINT: Do this first when F nand F are continuous and strictly increasing. The general case is obtained by smoothing F nand F by convoluting with a uniform distribution in
1.11. Let {f..Ln} be a sequence of p.m.'s such that for each f E CB, the sequence f31l f dlln converges; then f..Ln ~ f..L, where f..L is a p.m. [HINT: If the 4.5 UNIFORM INTEGRABILITY; CONVERGENCE OF MOMENTS I 99 hypothesis is strengthened to include every f in C, and convergence of real numbers
1.10. Find two sequences of p.m.' s {f..Ln} and {vn} such that J 1 ) J )or , but for no finite (a,b) is it true that f..Ln (a,b) -Vn (a,b) ---+ O.[HINT: Let f..Ln = orn , VII = OSn and choose {rn}, {sn} suitably.]
1.*9. The Levy distance of two s.d.f.'s F and G is defined to be the infimum of all €O > 0 satisfying the inequalities in (15). Prove that this is indeed a metric in the space of s.dJ. 's, and that F n converges to F in this metric if and only.~ D V D -'I roc d roo d l~ ?n ---+ ? anu Y_ocp-n ---+
1.8. If the r. v.' s X and Y satisfy for some €O, then their dJ.'s F and G satisfying the inequalities:(15) 'fix E ;)71 : F(x -E) -€O::s G(x) ::s F(x + E) + €o.Derive another proof of Theorem 4.4.5 from this.
1.7. Prove the Corollary to Theorem 4.4.4.
1.6. Let the r.v.'s {Xa} have the p.m.'s {f..La}. If for some real r> 0, eb { IX a I r} is bounded in ex , then {ga} is tight.
1.5. A set {gal of p.m.'s is tight if and only if the corresponding dJ.'s{F a} converge uniformly in ex as x ---+ -00 and as x ---+ +00.
1.* 4. Give an example to show that convergence in dist. does not imply that in pr. However, show that convergence to the unit mass oa does imply that in pr. to the constant a.
1.*3. Let f..Ln and f..L be as in Exercise 1. If the f n's are bounded continuous functions converging uniformly tof, then J f n df..Ln ---+ J f df..L.
1.2. Let f..Ln ~ f..L when the f..Ln'S are s.p.m.'s. Then for each f E C and each finite continuity interval I we have h f df..Ln ---+.0 f df..L.
1.1. Let f..Ln and f..L be p.m.'s such that f..Ln ~ f..L. Show that the conclusion in (2) need not hold if (a) f is bounded and Borel measurable and all f..Ln and f..L are absolutely continuous, or (b) f is continuous except at one point and every f..Ln is absolutely continuous. (To find even
1.9. Prove a convergence theorem in metric space that will include both Theorem 4.3.3 for p.m.'s and the analogue for real numbers given before the theorem. [HINT: Use Exercise 9 of Sec. 4.4.]
1.8. If!-Ln and f.l are p.m.'s and !-Ln(E) ---+ !-L(E) for every open set E, then this is also true for every Borel set. [HINT: Use (7) of Sec. 2.2.]
1.7. If g>n is a sequence of p.m. 's on (n, sr) such that gPn (E) converges for every E E :!ft, then the limit is a p.m. gPo Furthermore, if f is bounded and sr-measurable, then(The first assertion is the Vitali-Hahn-Saks theorem and rather deep, but it can be proved by reducing it to a problem of
1.6. Let {tin} be a sequence of finite measures on gJl. It is said to converge vaguely to a measure f..L iff (1) holds. The limit f..L is not necessarily a finite measure. But if f.1n (9[1) is bounded in 11, then f..t is finite.
1.5. Let {f n} be a sequence of functions increasing in .0/(1 and uniformly bounded there. sUPn.x If n (X)I < lvf < 00. Prove that there exists an increasing function [ on 9'21 and a subsequence {nd such that [n. (x) ---+ [(x) for every x. (This is a form of Theorem 4.3.3 frequently given; the
1.4. If a sequence of p.m.'s converges vaguely to an atomless p.m., then the convergence is uniform for all inter vals, finite or infinite. (This is due to P6Iya.)
1.3. Can a sequence of absolutely continuous p.m.' s converge vaguely to a discrete p.m.? Can a sequence of discrete p.m. 's converge vaguely to an absolutely continuous p.m.?
1.2. Prove that if (1) is true, then there exists a dense set D', such that f..Ln (I) ---+ f..L(I) where I may be any of the four intervals (a, b), (a, b], [a, b),[a, b] with a ED', bED'.
1.1. Perhaps the most logical approach to vague convergence is as follows.The sequence {f..Ln, n > I} of s.p.m.' s is said to converge vaguely iff there exists a dense subset D of gel such that for every a ED, bED, a I} converges. The definition given before implies this, of course, but prove the
1.20. Let {En} be arbitrary events satisfying(i) lim jI)(EIl )= 0, Il n then 9{limsuPIl Ell} = O. [This is due to Barndorff-Nielsen.]
1.19. If 2::n q!J(En) = 00 and lim{tt~(EjEk)} / {tqp (Ek)}2 = 1,/l j=! k=! k=!then ?fi{limsuPnEn} = 1. [HINT: Use (11) above.]84 I CONVERGENCE CONCEPTS
1.*18. Let {En} be events and {Ill} their indicators Prove the inequality Deduce from this that if (i) 2::n 9(En) = 00 and (ii) there exists c > 0 such that we have then 9{limsupEn} > O.11
1.17. If $(Xn) I and g(X~) is bounded in n, then 9{ lim Xn ~ I} > O. n-+oo[This is due to Kochen and Stone. Truncate Xn at A to Y n with A so large that f(Yn ) > 1 E for all n; then apply Exercise 6 of Sec. 3.2.]
1.*16. Strengthen Theorem 4.2.4 by proving that. In hm = 1 a.e. n-+oo g(l n)[HINT: Take a subsequence {kn} such that g(lnk) '" k2 ; prove the result first for this subsequence by estimating 9{lh -cff(lk) I > 8cf(ld}; the general case follows because if n k ::::; n < n k+! ,
1.15. If 2::n ~(IXn I > n) < 00, then. IXnl hm sup --::::; 1 a.e.n n
1.*14. If {Xn} is a sequence of independent r.v.'s with d.f.'s {Fn}, then:--':P{limn X/l = O} = 1 if and only if'iE > 0: 2::n {l -F n (E) + F/l (-E)} < 00.
1.13. If {Xn} is a sequence of independent and identically distributed r.v.'s not constant a.e., then go{XIl converges} = O.4.2 ALMOST SURE CONVERGENCE; BOREL-CANTELLI LEMMA I 83
1.* 12. Prove that the probability of convergence of a sequence of indepen dent r. v.' s is equal to zero or one.
1.11. Give a trivial example of dependent {E~} satisfying the hypothesis but not the conclusion of (6); give a less trivial example satisfying the hypoth esis but with 2P(lim sUPn En) = O. [HINT: Let En be the event that a real number in [0, 1] has its n-ary expansion begin with 0.]
1.10. Suppose that for a {Xn < a i.o. and X/1 > b i.o.} = 0;80 I CONVERGENCE CONCEPTS then limll ...... oc XIl exists a.e. but it may be infinite. [HINT: Consider all pairs of rational numbers (a.b) and take a union over them.]Under the assumption of independence, Theorem 4.2.1 has a striking
1.9. As in Exercise 8 show that 00 00 oo{1} {w: lim Xn(w) = O} =nun IXnl:::; - . n-+oo m m=1 k=ln=k
1.8. Let {Xn, n ~ 1} be any sequence of functions on n to 2/{1 and let C denote the set of 'v for which the numerical sequence {XI! ('0), n > 1}converges. Show that 00 00 00 c A U A A(m,n,n')m=ln=1 n'=1l+1 where/\(m, n, n') = w: max IXj(w) -Xk(w)1 :::;n< '
1.*7. {Xn} converges in pr. to X if and only if every subsequence{Xnk } contains a further subsequence that converges a.e. to X. [HINT: Use Theorem 4.1.5.]
1.6. Cauchy convergence of {X n} in pro (or in LP) implies the existence of an X (finite a,e,), such that Xn converges to X in pr. (or in LP). [HINT: Choose nk so that cf, Theorem 4.2.3.]
1.*5. If {Xn} converges on the set C, then for any f > 0, there exists Co C C with 9(C\Co) < f such that Xn converges uniformly in Co. [This is Egorov's theorem. We may suppose C = n and the limit to be zero. Let Fmk = n~=dw: I Xn(w):::; 11m}; then \1m, 3k(m) such that P?(F m.k(m)) > 1 -f/2m.Take
1.* 4. For any sequence of r. v.' s {X n} there exists a sequence of constants{An} such that X n IAn ---+ 0 a.e.
1.3. If {Xn} converges to a finite limit a.e., then for any E there exists M(E) < 00 such that !J'{sup IXn I ::::; M(E)} ~ 1 -E.1.4.2 ALMOST SURE CONVERGENCE; BOREL-CANTELLI LEMMA I 79
1.2. Let {Bn} be a countable collection of Borel sets in 'fl. If there exists a 8 > 0 such that m(Bn) ~ 8 for every n, then there is at least one point in .J1l that belongs to infinitely many Bn's.
1.1. Prove that 9(hmsupEn) ~ hm;o/>(Ell ), n n n Il
1.20. Let {XIl }be a sequence of independent r.v.' s with zero mean and unit variance. Prove that for any bounded r.V. Y we have limn-+oo
1.19. Let !1l(x) 1 + cos 21Tnx, f(x) 1 in [0, 1]. Then for each g E £1[0, 1] we have Z' fngdx -->l' fgdx, but ill does not converge to f in measure. [HINT: This is just the Riemann -Lebesgue lemma in Fourier series, but we have made f n ~ 0 to stress a POInt.]
1.18. If Xn"!'-X a.s., each Xn is integrable and infn J(Xn) > -00, then Xn ---+ X In L1.
1.17. Unlike convergence in pr., convergence a.e. is not expressible by means of metric. [HINT: Metric convergence has the property that if p(Xn , x) -1+ 0, then there eXIst E > 0 and {n k} such that p(xnk , x) ~ E for every k.]
1.*16. Convergence in pro for arbitrary r.v.'s may be reduced to that of bounded r. v.' s by the transformation X' = arctan X.In a space of uniformly bounded r.v.'s, convergence in pr. is equivalent to that in the metric Po(X, Y) = cf(IX -YI); this reduces to the definition given in Exercise 8 of
1.*15. Instead of the P in Theorem 4.1.5 one may define other metrics as follows. Let PI (X, Y) be the infimum of all E > 0 such that,~(IX -YI > E) ::::; E.Let P2(X, Y) be the infimum of ~{IX -YI > E} + E over all E > O. Prove that these are metrics and that convergence in pro is equivalent to
1.14. It is possible that for each w, limnXn (w) = +00, but there does not exist a subsequence {nk} and a set .6. of positive probability such that limk Xnk (w) = +00 on .6.. [HINT: On (1/, :13) define Xn (w) according to the nth digit of w.]
1.13. If sUPn Xn = +00 a.e., there need exist no subsequence {Xnk} that di verges to +00 in pr.
1.12. The sequence of extended-valued r.v. {Xn} is said to be bounded in pr. iff sUPn IXn I is bounded in pr.; {Xn} is said to diverge to +00 in pro iff for each lv! > 0 and E > 0 there exists a finite noCM, E) such that if n > no, then·~{IXnl > M} > 1 -E. Prove that if {Xn} diverges to +00 in pr.
1.11. The extended-valued r.V. X is said to be bounded in pro iff for each E > 0, there exists a finite M(E) such that 9{IXI :::; M(E)} :::: 1 -E. Prove that X is bounded in pro if and only if it is finite a.e.
1.*10. Let I be a continuous function on 0'(1. If Xn -+ X in pr., then I (X n) -+ I (X) in pr. The result is false if I is merely Borel measurable.[HINT: Truncate f at ±A for large A.]
1.9. Give an example in which cff(Xn) -+ 0 but there does not exist any subsequence {nk} -----+ 00 such that Xilk ----+ 0 in pr.
1.8. If Xn -+ X a e and Jin and ,II are the pm's of XI! and X, it does not follow that /J-n (P) -+ /J-(P) even for all intervals P.
1.7. If Xn -+ X in pr. and Xn -+ Y in pr., then X -Y a.e.
1.6. If Xn ----+ X, Yn ----+ Y, both in LP, then Xn ± Yn ---+ X ± Y in LfJ. If Xn -+ X in LP and Yn -+ Y in Lq, where p > 1 and l/p + l/q = 1, then
1.5. Convergence in LP implies that in LT for r < p.
1.* 4. Let I be a bounded uniformly continuous function in 0'(1. Then Xn -+ o in pr. implies {I (Xn)} -+ I (0). [Example:as in Theorem 4.1.5.]IXI I(x) = 1 + IXI
1.3. If Xn -+ X, Yn -+ Y both in pr., then Xn ± Yn -+ X ± Y, XnYn -+XY, all in pro
1.2. If 0 :::; Xn, Xn :::; X ELI and Xn -+ X in pr., then Xn -+ X in LI.
1.1. Xn -+ +00 a.e. if and only if \1M > 0: 2P{Xn < M i.o.} = O.
1.20. A typical application of Fubini' s theorem is as follows. If J is a Lebesgue measmable function of (x, y) such that f(x, y) 0 fOl each x E ~l and y rJ. Nx, where m(Nx) = 0 for each x, then we have also J(x, y) = 0 for each y rJ. N and x EN;, where m(N) = 0 and meN;) = 0 for each y rJ. N.
1.19. If J E .':01 X ~ and r r J J IJld(fYJ1 X ~) < 00, Q}xQ2 then
1.18. Prove that ill x ill =1= ill x ill, where ~ is the completion of g] with respect to the Lebesgue measure; similarly ill x .cJ3.

Showing 2000 - 2100 of 3340

First
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved