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elementary probability for applications

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

1.7. The convolution of two discrete distributions with exactly m and n atoms, respectIvely, has at least m + n -1 and at most mn atoms.
1.* 6. Prove that the convolution of two discrete dJ.' s is discrete; that of a continuous dJ. with any dJ. is continuous; that of an absolutely continuous d.f. with any dJ. is absolutely continuous.
1.5. If F 1 and F 2 are d.f.' s such that Fl = ~bl)aj jand F 2 has density p, show that F 1* F 2 has a density and find it.
1.4. Let Sn be as in (v) and suppose that Sn -+ Soo in pro Prove that 00 converges in the sense of infinite product for each t and is the chJ. of Soo.
1.3. Find the d.f. with the following ch.f.'s (a > 0, {3 > 0):a2 1 1 a2 + t2 ' (l -ait)f3' (l + a{3 -a{3eit )l/f3'[HINT: The second and third steps correspond respectively to the gamma and P6lya distributions.]
1.*2. Let feu, t) be a function on ( 00, (0) x ( 00, (0) such that for each u, f (u, .) is a ch.f. and for each t, f (-, t) is a continuous function; then[00 .J -00 feu, t) dG(u)is a ch.f. for any d.f. G. In particular, if f is a ch.f. such that limt~oo f(t)exists and G a d.f. with G(O) 0, then~OO
1.1 .. If f is a ch.f., and G a dJ. with G(O-) -0, then the following functions are all ch.f.' s:t f(ut) du, Jo[00 f(ut)e-U du, Jo rOO 10 f (ut) dG(u).
1.12. Let 9O{X1 = k} = Pb 1 < k < .e, L~=1 Pk = 1. Let N(n, w) be the number of values of j, 1 < j < n, for which X} k and Prove that lI--+OC n1 lim -log IICn , w) exists a.e.and find the limit. [This is from information theory.]
1.* 11. Let 1 be continuous and belong to Lr (0, (0) for some r > 1, and g()..) = [00 e-At 1(t) dt.Jo Then f(x)n-->oo (n -1)!( _1)n-1 lim (:)ng(n_l)(:), where gCn-l) is the (n -l)st derivative of g, unifonnly in every finite interval.[HINT: Let ), > 0, JP{X1 (,l .. ) < t} -1 e-At . Then and
1.10. Let r be a positive integer-valued r. v. that is independent of the Xn 's.Suppose that both r and X 1 have finite second moments, then a-2(Sr) = e''(r)a-2(XJ) + a-2 (r)(ct(X1))2.
1.*9. In Exercise 7, find the dJ. of XIJ(t) for a given t. (g'{Xv(t)} is the mean lifespan of the object living at the epoch t; should it not be the same as t {X 1}, the mean lifespan of the given species? [This is one of the best examples of the use or misuse of intuition in probability theory.]
1.8. Theorem 5.5.3 remains true if J(X1) is defined, possibly +00 or -00.
1.*7. Consider the special case of renewal where the r.v.'s are Bernoullian taking the values 1 and 0 with probabilities p and 1 -p, where 0 < p < 1.Find explicitly the dJ. of v(o) as defined in Exercise 6, and hence of vet)for every t> 0. Find {{vet)} and {{v(t)2}. Relate v(t,w) to the N(t,w) in
1.*6. POI each t > 0, define o(t,(O) min{n.ISn(w)1 > t}if such an n exists, or +00 if not. If gp(X 1 =1= 0) > 0, then for every t > 0 and r> 0 we have .:.?-'{ vet) > n} ~ An for some A < 1 and all large n; consequently cf {v(tY} < 00. This implies the corollary of Theorem 5.5.2 without recourse to
1.5. If {,(Xl) > 0, then,l,i':'oo PI>{Q [Sn :5 III = o.
1.*4. Let Sn and N(t) be as m Theorem 5.5.2. Show that 00 e?{N(t)} = Lq>{Sn ~ t}.n=l'IliiS remams true If X 1 takes both pOSItIve and negatIve values.
1.3. Find the distribution of Yllk , ] < k < n, in (1) [These r v.'s are called order statistics.]
1.*2. Let F n and F be as in Theorem 5.5.1; then the distribution of sup IF n (x, (0) F(x)1 -oo
1.1. Show that equality can hold somewhere in (1) with strictly positive probability if and only if the discrete part of F does not vanish.
1.13. Under the assumptions in Theorem 5.4.2, if Sn/n converges a.e. then/ (IXII) < 00. [Hint: Xnln converges to° a.e , hence s:P{ I Kill> 11 i a } 0;use Theorem 4.2.4 to get Ln :-1'{ IX 11 > n} < 00.]
1.12. If [(Kd =1= 0, then max}
1.11. Prove the second alternatIve m Theorem 5.4.3.
1.10. Suppose there exist ana, 0 < a < 2, a =1= 1, and two constants A and A2 such that 1'rIn, 'rIx > 0: Al A2 -::::: 0O{IXnl > x} < -. xa -xa If a > 1 , suppose also that f (Xn ) o for each n Then for any sequence {an}increasing to infinity, we have[ThI~ ~esult, due to P. Levy and MarcmkiewIcz,
1.9. Construct an example where I(X+) = "'(X-)-+00 [ L 1 (r 1 -+00 and S In ~ a.e. Hll\7: e6 0 < a < fJ < 1 and take adJ. F such that 1 _ F (X;I,......, x-a as x ~ 00, and J-oo Ixltl dF(x) < 00. Show that L jj){ m~x X-!-::::: n Ila'} < 00 n I::::J::::n J for every a' > a and use Exercise 3 for ~n
1. *8. If (rCIXII) < 00, then the sequence II n IS um S nl n ~ (; (X I) in L I as well as a.e.
1.7. We have SII/n ~ 0 a.e. if and only if the following two conditions are satisfied:(i) Sn/n ~ a in pr.,(ii) S 2" 12n ~ 0 a.e.;an alternative set of conditions is (i) and(iii) 'VE > 0: LII ?J>(IS2"+' -S2"! > 21lE) < 00.{S I }. . fonnly integrable and
1.6. Let 0'(X 1 ) -0 and {ell} be a bounded sequence of real numbers. Then 1 n[HINT: Truncate XII at n and proceed as in Theorem 5.4.2.]
1.5. Let Xn take the values ±n8 with probability ~ eaeh. If 0 .::::: 0 ~? [HINT: To answer the question, use theorem 5.2.3 or Exercise 12 of Sec. 5.2; an alternative method is to consider the characteristic function of S,dn (see Chapter 6).]
1.4. Both Theorem 5.4.1 and its complement in Exercise 2 above are "best possible" in the following sense. Let {all} and 0, Then there exists a sequence of independent and identically distributed r.v.'s{Xn} such that [(Xn) = 0, $( an} = 00 by letting each Xn take two or three values only according
1.3. Let {Xn} be independent and identically distributed r.v.'s such that!(IXIIP) < 00 for some p:O < p < 2; in case p> 1, we assume also that/ (X I ) = O. Then S n n -(1/ P )-E ~ 0 a.e. For p = 1 the result is weaker than Theorem 5.4.2.
1.*2. There is a complement to Theorem 5.4.1 as follows. Let {an} and
1.*1. If £(Xi) = +00, leX}) < 00, then Snln ~ +00 a.e.
1.*10. If Ln ±Xn converges a.e. for all choices of ±1, where the Xn's are arbitrary r.v.'s, then Ln Xn 2 converges a.e. [HINT: Consider Ln rn (t)Xn (w)where the r,1 's are coin-tossing r v's and apply Fubinj's theorem to the space of (t, w).]
1.*9. Let {X n} be independent and identically distributed, taking the values o and 2 with probability ~ each; then converges a.e. Prove that the limit has the Cantor d.t. discussed In Sec. 1.3.Do Exercise 11 in that section again; it is easier now.
1.8. Let {Xn}, where n = 0, ±1, ±2, ... , be independent and identically distributed according to the normal distribution
1.7. For arbitrary {Xn}, if L cf(IXn I) < 00, n then LII Xn converges absolutely a.e.
1.* 6. The following analogue of the inequalities of Kolmogorov and Otta viani is due to P. Levy. Let Sn be the sum of n independent r.v.'s and S~ = Sn -mo(Sn), where mo(Sn) is a median of Sn' Then we have 2p{m~x ISJI > E}:S 3gp{IS~1 > ~}. lS;.JS;.n 2[HINT: Try "4" in place of "3" on the right.]
1.*5. But neIther Theorem 5.3.2 nor the alternatIve indIcated In the prece ding exercise is necessary; what we need is merely the following result, which is an easy consequence of a general theorem in Chapter 7. Let {X n} be a sequence of independent and uniformly bounded I. v.' s with 0 2 (8 n)
1.4. Let {XI!, X~ , n > I} be independent r v ' s sJ]ch that Xn and X~ have the same distribution. Suppose further that all these r.v.'s are bounded by the same constant A. Then nconverges a.e. if and only if nlIse Exercise 3 to prove this without recourse to Theorem 5.3 3, and so finish the
1.3. Theorem 5.3.2 has the following companion, which is easier to prove.Under the joint hypotheses in Theorems 5.3.1 and 5.3.2, we have(A + E)2 .9'{ max IS'I < E} < ---J --(J2 (Sn) . lS;.jS;.n
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.
1.11. If I is completely monotonic in (0, 00) with 1(0+) = +00, then I is the Laplace transform of an infinite measure fJ., on .9i\:I(A) = r e-AxfJ.,(dx). }[JI{+[HINT: Show that F n (x) :s e2x81(8) for each 8 > 0 and all large n, where Fn is defined in (7). Alternatively, apply Theorem 6.6.4 to I(A
1.*10. If I > 0 on (0,00) and has a derivative I' that is completely mono tonic there, then 1/1 is also completely monotonic.
1.9. Given a function CJ on !Ji>+ that is finite, continuous, and decreasing to zero at infinity, find a CJ-finite measure fJ. on :~+ such that Vt > 0:r CJ(t -s)fJ.(ds) = 1.J[O.t][HINT: Assume CJ(O) = 1 and consider the dJ. 1 -CJ.]
1.*8. In the notation of Exercise 7, prove that for every A, fJ. E .~+:A A!C).) !CM)·
1.7. Let leA) = Jooo e-A.x f(x)dx where fELl (0, (0). Suppose that f has a finite right-hand derivative l' (0) at the origin, then A fCO) lim kfCk), f'(O) = lim A[Af(A) -f(O)].).,--+00
1.6. Let F be an s.dJ. with support in f?I4. Define Go F, for 11 2: 1. Find Gil (A) in terms of F(A).
1.5. Use Exercise 3 to prove that for any d.f. whose support is in a finite interval, the moment problem is determinate.
1.*4. Let {FIl }be s.d.f.'s. If AO > 0 and limn~ooFn (A) exists for all A 2: AO, then {Ell} converges vaguely to an s d f
1.*3. Let F and G be s.d.f.'s. If AO > 0 and F(A) = G(A) for all A ~ AO, then F = G. More generally, if F(nAo) = G(nAo) for integer n ~ 1, then F = G.[HINT: In order to apply the Stone-Weierstrass theorem as cited, adjoin the constant 1 to the family {e-A.x, A ~ AO}; show that a function in Co can
1.* 2. Two uncorrelated r. v.' s with a joint normal d.f. of arbitrary parameters are independent. Extend this to any finite number of r.v.'s.
1.1. If X and Yare independent r. v.' s with normal d.f.' s of the same variance, then X + Y and X -Y are independent.
1.9. Show that in Theorem 6.3.2, the hypothesis (a) cannot be relaxed to require convergence of {In} only in a finite interval It I :::; T.
1.8. Suppose I(t, u) is a function on gr2 such that for each u, 1(·, u) is a ch.f.~ and for each t, I(t,·) is continuous. Then for any d.f. G, exp{I: [/(t, u) -l]dG(U)}is a ch.f.
1.7. Construct a ch.f. that vanishes in [-b, -a] and [a, b], where 0 < a
1.6. Show that there is a ch.f. that has period 2m, m an integer ~ 1, and that is equal to 1 -It I in [-1, +1]. [HINT: Compute the Fourier series of such a function to show that the coefficients are positive.]
1.*5. Another proof of Theorem 6.5.3 is as follows. ShO'.v that 70 tdj'(t) = 1 and define the d.f. G on .0'C+ by G(u)r t df' (t)N ext show that Hence if we set rill) f (u, t) = 1 --;; V 0(see Exercise 2 of Sec 6 2), then f(t) =r feu, t)dG(u).J[O,OO)Now apply Exercise 2 of Sec. 6.1.
1.4. If F is a symmetric stable distribution of exponenta, 0 < a < 2, then J~oo Ixlr dF(x) < 00 for r < a and = 00 for r ::::a. [HINT: Use Exercises 7 and 8 of Sec. 6.4.]
1.3. If {X n} are independent r. v.' s with the same stable distribution of exponenta, then I:Z=l Xdn Ija has the same distribution. [This is the origin of the name "stable".]
1.2. Show that the following functions are ch.f.' s:1 1 + It I ' f(t) = { 1 -Itla, 0,{I -Itl, f(t) = _1 41tl'if It I :s 1;if It I :::: 1;if 0< It I :s!;if It I ~ !.0< a :s 1,
1.1. If f is continuous in ~1 and satisfies (3) for each x E .~1 and each T > 0, then f is positive definite.
1.35. Random variables defined on ® are also referred to as defined"modulo I". The theory of addition of independent r.v.'s on ® is somewhat simpler than on ,921, as exemplified by the following theorem. Let {X j, j ~ I}be independent and identically distributed r. v.' s on ® and let S k = L~= 1
1.34. Suppose that the space V is replaced by its closure [0, 1] and the two points° and 1 are identified; in other words, suppose '1/ is regarded as the circumference ® of a circle; then we can improve the result in Exercise 33 as follows. If there exists a function g on the integers such that f
1.*33. f.-Lk~f.-L if and only if I!.ik(·) ~ I p.(.) everywhere.
1.*32. f.-L is equidistributed on the set {jn-I ,° :s j :s n -I} if and only if I p.(j) = ° or 1 according to j f n or j In.
1.31. Prove that If p.(n )1 -I If and only If fJ, has its support in the set{8o + jn-1 ,° :s j :s n -I} for some 80 in (0, n-I ].
1.30. EstablIsh the InVerSIOn formula expressIng f.-L In terms of the f p.(n)'s.Deduce again the uniqueness result in Exercise 29. [HINT: Find the Fourier series of the indicator function of an interval contained in 0t/.]
1.29. Define for each n:f (n \ = [ e27rinx u(dx\ ~t( ) Jw ""')Prove by WeIerstrass ' s approxImatIOn theorem (by tngonometncal polyno mials) that if 11*, (n) = 11*2 (n) for every n > 1 , then J.-L 1 = U2. The conclusion becomes false if 11 is replaced by [0, 1].
1.28. Let Qn be the concentration function of S n = 2:J= 1 X j, where the x /s are independent r.v.'s having a common nondegenerate dJ. F. Then for every h > 0,[HINT: Use Exercise 27 above and Exercise 16 of Sec. 6.1. This result is due to Levy and Doeblin, but the proof is due to Rosen.]In
1.27. If I is any nondegenerate ch. I, then there exist constants A >°and 8 >° such that II (t)1 :s 1 -At2 for It I :s 8.[HINT: Reduce to the case where the d.f. has zero mean and finite variance by translating and truncating.]
1.26. If F is symmetric and J Ixl dF(x) < 00, then n.9>{Sn = j} ~ 00.[HINT: 1 -l(t) = o(ltl) as t ~ 0.]
1.25. If F #-8o, then there exists a constant A such that for every j::?>{Sn = j} :s An -1/2.[HINT: Use a special case of Exercise 27 below.]
1.*24. If J x dF(x) = 0, J x2 dF(x) = (J2, then for each integer j:1 n 1/2~}J{SIl = j} ~ -Jfii'(J 2Jr[HINT: Proceed as in Theorem 6.4.4, but use (15).]
1.*23. If {Xn} is a sequence of independent and identically distributed r.v.'s, then there does not exist a sequence of constants {en} such that Ln (Xn -en)converaes a e lInless the common d f is degenerate b , In Exercises 24 to 26, let Sn = LJ-l Xj, where the Xjs are independent r.v.'s with a
1.22. Let f(s, t) be the ch.f. of a 2-dimensional p.m. v. If If(so, to)1 = 1 for some (s-o, to) #-(0,0), what can one say about the support of v?

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