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elementary probability for applications

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

1.*17. For arbitrary events {Ej , I :::::: j :::::: n}, we have 11:::: L?~(Ej) -L ~:Jj)(EjEk)'j=1 I~j
1.16. Generalize (14) to.Jl(C) = :-P«C I WI, ... , wdY11 (dwI)" . .Yk(dwd e(1 k)J~ .l>((C I tv), ... , tvd);-Y>(dtv), where C (1, ... , k) is the set of (WI, ... , Wk) which appears as the first k coordinates of at least one point in C and in the second equation WI, ... , Wk are regarded as
1.15. Modify Example 4 further so as to allow En to take the values 1 and ° with probabilities PI! and I -PI1' where° < Pn < I, but Pn depends on n.
1.* 14. Modify Example 4 so that to each x in [0, I] there corresponds a sequence of independent and identically distributed r.v.'s {En, n > I}, each taking the values 1 and 0 with probabilities p and 1 p, respectively, where ° < P < 1. Such a sequence will be referred to as a "coin-tossing game
1.13. Gener alize Example 4 by considering any s-ary expansion where s >2 is an integer:x 11=1 where En Sl! '0, 1, ... , s 1.
1.12. The r. v.' s {E j} in Example 4 are related to the "Rademacher func tions":What is the precise relation?
1.11. If X and Y are independent, (f,( IX I P) < 00 for some P > I, and cf(Y) = 0, then I(IX + YIP) > g(IXIP). [This is a case of Theorem 9.3.2;but try a direct proof!]
1.*10. If X and Y are independent and for some P > 0: ~(IX + YIP) < 00, then (,l,'(IXIP) < 00 and cf(IYIP) < 00.
1.*9. If X and Y are independent and £'(X) exists, then for any Borel set B, we have I x d?JJ = I (X).O/(Y E B).{YEB}
1.8. Let {X), 1 :'S j :'S n} be independent with d.f.' s {F), 1 :'S j :'S n}. Find the dJ. of max) X) and min) X).66 I RANDOM VARIABLE. EXPECTATION. INDEPENDENCE
1.7. If {EJ , 1 < j < oo} are independent events then)=1 )=1 where the infinite product is defined to be the obvious limit; similarly
1.6. The r.v. X is independent of itself if and only if it is constant with probability one. Can X and f (X) be independent where f E 2(31?
1.5. If {Xa} is a family of independent r.v.'s, then the B.F.'s generated by disjoint subfamilies are independent. [Theorem 3.3.2 is a corollary to this proposition. ]
1.4. Fields or B F 's .vfa(C .C0 ) of any family are said to be independent iff any collection of events, one from each g;, forms a set of independent events.Let :¥aD be a field generating :¥a. Prove that if the fields ~ are independent, then so are the B.F.'s ;:-'#0" Is the same conclusion true
1.*3. If the events {EO', ex E A} are independent, then so are the events{Fa, ex E A}, where each Fa may be EO' or E~; also if {Ali, f3 E B}, where B is an arbitrary index set, is a collection of disjoint countable subsets of A, then the events are independent.
1.*2. If XI and X2 are independent r.v.'s each assuming the values +1 and -1 with probability ~, then the three r.v.'s {XI,X2,XIX2} are pairwise independent but not totally independent. Find an example of n r.v.'s such that every n - 1 of them are independent but not all of them. Find an example
1.20. For r > ], we have roo 1 r I[HINT: By Exercise 17, ur r I(X!\ ur ) = 10 ./I(X > x)dx = }o0"(XI / r > v)rvr-1 dv, substitute and invert the order of the repeated integrations.]t not according to another!
1.*19. If {XII} is a sequence of identically distributed r.v.'s with finite mean, then 1 lim -t'{ max IX il} = O.n 11 IS}SII[HINT: Use Exercise 17 to express the mean of the maxImum.]
1.18. Prove that J~oo Ixl dF(x) < 00 if and only if[0 F(x) dx < 00 and [00 [1 _ F(x)] dx < 00. Jo
1.17. If F is a dJ. such that F(O-) = 0, then roo [1 F(x)} dx 10 }~oo xdF(x) .s: +00.Thus if X is a positive r.v., then we have!'(X) = [00 .?>{X > x}dx = [00 .?>{X > x}dx.)0 Jo
1.* 16. For any dJ. and any a ~ 0, we have I: [F(x +a) -F(x)] dx = a.
1.15. If p > 0, e'(IXIP) < 00, then xP~{IXI > x} = 0(1) as x --+ 00.Conversely, if xP3l{IXI > x} = 0(1), then (f'(IXIP-E) < 00 for 0 < E < p.
1.*14. If p > 1, we have and so P 1 n 1 n -"'X· :s -"'IX·IP n0 } n0 }j=l j=l 52 I RANDOM VARIABLE. EXPECTATION. INDEPENDENCE we have also Compare the inequalities.
1.* 13. If Xj > 0, then nor >L cff(Xj)j=l according as p :::: 1 or p 2: 1.
1.*12. If X> 0 and Y> 0, p > 0, then rS'{(X + y)P} < 2P{t(XP) +e(yP)}. If p > 1, the factor 2P may be replaced by 2P-1 . If 0 :::: p :::: 1, it may be replaced by I.
1.10. Prove that If 0 :s r < rl and &'(IXI") < 00, then cf(IXlr ) < 00. Also that {(IXlr ) < 00 if and only if cS,(IX -air) < 00 for every a.for a :s x :s I.
1.9. Another proof of (14): verify it first for simple r.v.'s and then use Exercise 7 of Sec. 3.2.
1.*8. For any two sets Al and A2 in qT, define then p is a pseudo-metric in the space of sets in :F; call the resulting metric space M(0T, g"». PIOve that fOl each integrable r .v. X the mapping of MC~ , go)to ll?l given by A -+ .~ X dq? is continuous. Similarly, the mappmgs on M(::f , ;:7J1) x
1.*7. Given the r.v. X with finite {(X), and E > 0, there exists a simple r.v.X E (see the end of Sec. 3.1) such that cF (IX - X E I) < E.Hence there exists a sequence of simple r. v.' s X m such that lim cf(IX -Xml) = O.m-+OO We can choose {Xm} so that IXml :s IXI for all m.
1.* 6. Suppose that supn IXIl I :s Y on A with fA Y d:~ < 00. Deduce from Fatou's lemma: j (lim X,Jd~ ~ lim r Xn d?J1.A 11-+00 11-+00 J A Show this is false if the condition involving Y is omitted.
1.4. Let c be a fixed constant, c > O. Then c&'(IXI) < 00 if and only if L~(IXI :::: cn) < 00.11-1 In particular, if the last series converges for one value ofc, it converges for all values of c.5. For any r> 0, J'(IXI') < 00 if and only if 00 L n,-l ~(IXI :::: n) < 00.n=l 3.2 PROPERTIES OF
1.3. Let X:::: 0 and JnX d9 = A, 0 < A < 00. Then the set function v defined on:if as follows:1 r v(A) =J X d?7J, A A is a probability measure on 4
1.*2. If {(IXI) < 00 and limn...-+oo 9(An) 0, then limn...-+oo fAX d9 O. n In particular lim [ X d9= O.
1.1. If X:::: 0 a.e. on A and JA X dPJ5 = 0, then X = a a.e. on A.
1.12. Generalize the assertion in Exercise 11 to a finite set of r.v.'s. [It is possible to generalize even to an arbitrary set of r.v.'s.]
1.*11. Let ':!ft {X} be the minimal B.F. with respect to which X is measurable.Show that A E ~ {X} if and only if A = X I (B) for some B E qjl. Is this B unique? Can there be a set A rt. /231 such that A -X-I (A)?
1.9. If f is Borel measurable, and X and Y are identically distributed, then so are f(X) and fEY).lim sup An, lim inf An in terms of those of A I, A2, or An. [For the definitions of the limits see Sec. 4.2.]
1.8. If Q is discrete (countable), then every r.v. is discrete. Conversely, every r.v. in a probability space is discrete if and only if the p.m. is atomic.[HINT: Use Exercise 23 of Sec. 2.2.]
1.7. The sum, difference, product, or quotient (denominator nonvanishing)of the two discrete r. v.' s is discrete.
1.6. Is the range of an r.v. necessarily Borel or Lebesgue measurable?
1.*5. Suppose X has the continuous dJ. F, then F(X) has the uniform distribution on [0,1]. What if F is not continuous?
1.*4. Let e be uniformly distributed on [0,1]. For each dJ. F, define G(y) =sup{x: F(x) :s y}. Then G(e) has the dJ. F.
1.*3. Given any p.m. fJ., on C::f?I, ::i'3 I ), define an r.v. whose p.m. is fJ.,. Can this be done in an arbitrary probability space?
1.2. If two r.v.'s are equal a.e., then they have the same p.m.
1.1. Prove Theorem 3.1.1. For the "direct mapping" X, which of these properties of X-I holds?3.2 PROPERTIES OF MATHEMATICAL EXPECTATION I 41
1.*25. Let f be measurable with respect to ~, and Z be contained in a null set. Define f={~ on ZC, on Z, where K is a constant. Then f is measurable with respect to ~ provided that(Q, ~ , gP) is complete. Show that the conclusion may be false otherwise.
1.*24. A point x is said to be in the support of a measure JL on 273n iff every open neighborhood of x has strictly positive measure. The set of all such points is called the support of JL. Prove that the support is a closed set whose complement is the maximal open set on which JL vanishes. Show
1.23. Prove that if the p.m. g; is atomless. then given any a in [0. 1]there exists a set E E !,h with ::P(E) =a. [HINT: Prove first that there exists E with "arbitrarily small" plObability. A quick plOof then follows flOm Zom's lemma by considering a maximal collection of disjoint sets, the sum of
1.22. For an arbitrary measure €7' on a B.F. tf, a set E in g; is called an atom of ,q; iff ,CiJi(E) >° and FeE, F E;!/r imply 9(F) = £:?P(E) or £:?P(F) =0. ~j"J is called atomic iff its value is zero over any set in ?7 that is disjoint from all the atoms. Prove that for a measure M on
1.21. Suppose that F has all the defining properties of a dJ. except that it is not assumed to be right continuous. Show that Theorem 2.2.2 and Lemma remain valid with F replaced by ft, provided that we replace F(x), F(b), F(a)in (4) and (5) by F(x±), F(b± ). F(a± ), respectively. What
1.*20. Let (Q,;!f, £:?P) be a probability space and .'1+1 a Borel sub field of;:",h. Prove that there exists a minimal B.F. 9
1.19. Let L I" be as in the proof of Theorem 2.2.5 and A6 be the set of all null sets in (Q, ;!f , .:?P). Then both these collections are monotone classes, and closed with respect to the operation "\".
1.18. Show that the;!f in (9) I is also the collection of sets of the form FUN [or F\N] where F E q; and N E ~,V.
1.17. Show that Theorem 2.2.3 may be false for measures that are O"-finite on ;:",h. [HINT: Take Q to be {I, 2, ... , oo} and st to be the finite sets excluding 00 and their complements, JL(E) = number of points in E, JL(oo) =1= v(oo).]
1.16. Show by a trivial example that Theorem 2.2.3 becomes false if the field ~ is replaced by an arbitrary collection that generates 87.
1.* 15. Translate the construction of a singular continuous d.f. in Sec. 1.3 in terms of measures. [It becomes clearer and easier to describe!] Generalize the construction by replacing the Cantor set with any perfect set of Borel measure zero. What if the latter has positive measure? Describe a
1.14. Translate Theorem 1.3.2 in terms of measures.
1.13. fJ, is called singular iff there exists a set Z with m(Z) = 0 such that fJ,(ZC) = O. This is the case if and only if F is singular. [HINT: One ha1f is proved by using Theorems 1.3.1 and 2.1.2 to get IB F' (x) dx ~ fJ,(B) for B E
1.12. fJ, is called atomic iff its value is zero on any set not containing any atom. This is the case if and only if F is discrete. fJ, is without any atom or atomless if and only if F is continuous.
1.*11. An atom of any measure g on gel is a singleton {x} such that fJ,({x}) > O. The number of atoms of any O"-finite measure is countable. For each x we have p;({x}) F(x) F(x ).
1.10. Instead of requiring that the E/s be pairwise disjoint, we may make the blOader assumption that each of them intersects only a finite number in the collection. Carry through the rest of the problem.
1.*9. Let {ff be a countable collection of pairwise disjoint subsets {E j, j > 1}
1.8. gj1 contains every singleton, countable set, open set, closed set, Go set, F (J set. (For the last two kinds of sets see, e.g., Natanson [3].)
1.7. The B F .q(11 on @l is also generated by the class of all open intervals or all closed intervals, or all half-lines of the form (-00, a] or (a, (0), or these intervals with rational endpoints. But it is not generated by all the singletons of gel nor by any finite collection of subsets of gel.
1.* 6. Now let I!!.. t/: gT be such that I!!.. C F E ?7 =} P?(F) = 1.Such a set is called thick in (Q, .~, ~). If E = I!!.. n F, F E ?7, define :?/J*(E) =:?/J(F). Then :?/J* is a well-defined (what does it mean?) p.m. on (I!!.., I!!.. n 5T).This procedure is called the adjunction of n to (ft, 'ff,
1.5. Prove that the trace of a B.F. ;:"/1 on any subset I!!.. of Q is a B.F.Prove that the trace of (Q, ;:",,? , 0") on any I!!.. in;}J is a probability space, if.o/>(I!!..) > O.
1.4. Prove the nonexistence of a p.m. on (Q, .1'), where (Q, J) is as in Example 1, such that the probability of each singleton has the same value.Hence criticize a sentence such as: "Choose an integer at random".
1.3. In the preceding example show that for each real number a in [0, 1]there is an E in (7 such that ??(E) =a. Is the set of all primes in ? ? Give an example of E that is not in {f.
1.* 2. Let Q be the space of natural numbers. For each E C Q let N n (E)be the cardinality of the set E I I [0, n] and let {{ be the collection of E's for which the following limit exists:l ' Nil (E) 1m .Il~OO n,7> is finitely additive on (? and is called the "asymptotic density" of E. Let E ={all
1.1. For any countably infinite set Q, the collection of its finite subsets and their complements forms a field ?ft. If we define ~(E) on q; to be 0 or 1 according as E is finite or not, then qp is finitely additive but not countably so.
1.12. Let {i' be a M.e. of subsets of llln (or a separable metric space)containing all the open sets and closed sets. Prove that (i' :) qjn (the topological Borel field defined in Exercise 11). [HINT: Show that the minimal such class is a field.]
1.11. Take n = };?Il or a separable metric space in Exercise 10 and let 9 be the class of all open sets. Let ;Y(' be a class of real-valued functions on n satisfying the following conditions.(a) 1 E ;/(' and 1D E ,/(' for each D E ~;(b) dr' is 'a vector space, namely: if fIE ;Yr', f 2 E ,'I(' and
1.10. Let Q be a class of subsets of Q having the closure property (in);let ,I:I be a class of sets containing n as well as ~, and having the closure properties (vi) and (x). Then sf contains the B.P. generated by fZ. (This is Dynkin's form of a monotone class theorem which is expedient fO! certain
1.9. If:Yr is a B.P. generated by a countable collection of diSjoint sets{All}, such that Un All = n, then each member of :1t is just the union of a countable subcollection of these Ail's.
1.*8. Let?T be a B.P. generated by an arbitrary collection of sets {Een ex E A}. Prove that for each E E ~, there exists a countable subcollection {Eaj' j >I} (depending on E) such that E belongs already to the B.F. generated by thIS subcollection. [HINT: Consider the class of all sets with the
1.7. A B.P. is said to be countably generated iff it is generated by a count able collection of sets. Prove that if each '2fj is countably generated, then so is \/00 ~ "J I~J'
1.* 6. The union of a countable collection of B.P.' s {~j} such that q; c ~j+l need not be a B.P., but there is a minimal B.P. containing all of them, denoted by v j ;;'fj. In general vaEA~a denotes the minimal B.P. containing all s+a, ex EA.[HINT: n = the set of positive integers; ~fj = the B.P.
1.5. The intersection of any collection of B.P.'s {.':"fa, ex E A} is the maximal B.P. contained in all of them; it is indifferently denoted by naEA ~ or AaEA.':"fa.
1.4. If n is countable, then J is generated by the singletons, and conversely. [HINT: All countable subsets of n and their complements form a B.P.]
1.3. If n has exactly 11 points, then J has 211 members. The B.P. generated by 11 given sets "without relations among them" has 2211 members.
1.*2. The best way to define the symmetric difference is through indicators of sets as follows:(mod 2)where we have arithmetical addition modulo 2 on the right side. All properties of t,. follow easily from this definition, some of which are rather tedious to verify otherwise. As examples:(A t,. B)
1.*1. (UjAj)\(UjBj) C U/Aj\Bj).(njAj)\(njBj) c U/Aj\Bj). When i£ there equality?
1.* 15. The Cantor d.f. F is a good building block of "pathological"examples. For example, let H be the inverse of the homeomorphic map of [0,1]onto itself: x ---7 ~ [F(x) + x]; and E a subset of [0,1] which is not Lebe£gue measurable. Show that where H(E) is the image of E, IB is the indicator
1.14. Given any closed set C in (-00, +(0), there exists a d.f. whose support IS exactlyc. [HINT: Such a problem becomes easier when the corre sponding measure is considered; see Sec 2 2 below]
1.*13. Consider F on [0,1]. Modify its inverse F-1 suitably to make it smgle-valued m [0,1]. Show that F 1 so modified is a discrete dJ. and find its points of jump and their sizes
1.12. Extend the function F on [0,1] trivially to (-00, (0). Let {Tn} be an enumeration of the rationals and 00 1 G(x) = ~ -FCrn + x). ~2n n=l Show that G IS a d.f. that is strictly increasing for all x and singular. Thus we have a singular d f with support ( 00, 00)
1.11. Calculate 10' xdF(x), 10' x2 dF(x), 10' ej,x dF(x).[HINT: This can be done directly or by using Exercise 10; for a third method see Exercise 9 of Sec. 5.3.]
1.10. For each x E [0, 1], we have 2F (~) = F(x), 2F (~+~) -1 = F(x).
1.*9. It is well known that any point x in C has a ternary expansion without the digit 1:Prove that for this x we have all = 0 or 2.00~ an F(x) = ~ 211+1 .n=1 1.3 ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS I 15
1.Prove that the support of F is exactly C.
1.7. Prove that a singular function as defined here is (Lebesgue) measurable but need not be of bounded variation even locally. [HINT: Such a function is continuous except on a set of Lebesgue measure zero; use the completeness of the Lebesgue measure.]
1.6. Prove that a discrete distribution is singular. [Cf. Exercise 13 of Sec. 2.2.]
1.5. Under the conditions in the preceding exercise, the support of F is the closure of the set {t I l(t> > OJ; the complement of the support is the interior of the set {t I I(t) = OJ.
1.* 4. Suppose that F is a d.f. and (3) holds with a continuous I. Then F' = I ~ 0 everywhere.
1.* 3. If the support of a d.f. (see Exercise 6 of Sec. 1.2) is of measure zero, then F is singular. The converse is false.1.3 ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS I 13
1.2. Prove Theorem 1.3.2.
1.1. A d.f. F is singular if and only if F = Fs; it is absolutely continuous if and only if F = Fac.
1.7. Prove that the support of any d.f. is a closed set, and the support of any continuous d.f. is a perfect set.
1.* 6. A point x is said to belong to the support of the d.f. F iff for every E > 0 we have F(x + E) -F(x -E) > O. The set of all such x is called the support of F. Show that each point of jump belongs to the support, and that each isolated point of the support is a point of jump. Give an example
1.5. Theorem 1.2.2 can be generalized to any bounded increasing function.More generally, let f be the difference of two bounded increasing functions on( 00, +(0); such a function is said to be of bounded Val iatioll there. Define its purely discontinuous and continuous parts and prove the

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