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elementary probability for applications

Probability Theory Independence Interchangeability Martingales 3rd Edition Yuan Shih Chow, Henry Teicher - Solutions

Let Q = (0,11. Fclass of subsets of Q. !!!)[0]) and define M. 21, and (F)(FX) Prove the following (i) Each point of 2 is an set of p-measure zero. (u) The set of all points of with only finitely many coordinates equal to 1 has p-measure () Define --D.- FAA), AF(iv) For each ere, z is a 1-1 map of
are known as the Rademacher functions.
In Exercise 2, define Z, Z)-1 or-1 according as the integer i for which (1-1/2
Let (v) be the probability space consisting of Lebesgue measure on the Borel subsets of [0, 1]. Each in [0, 1] may be written in binary expansion as o-X,X, - where X-X)-0 or 1 and this expansion is unique except for a set of as of the form w/2" which is countable and hence has probability (Lebesgue
Verify that it is possible to define a sequence of independent rvs (X) with specified dfs F, on a probability space.
Random variables (X,. Isis) are called interchangeable if their joint df. is a symmetric function, ie, invariant under permutations, and evs (X1) are interchangeable if every finite subset is interchangeable. Prove that if (X1) are, interchangeable rvs, then p(X, X20 Hint: X20
Random variables XX, have a multinomial distribution if PIX,-x, s kx, for any nonnegative integers x Isisk, with and zero otherwise. Here, n is a positive integer and Pla Isisk Prove that if (4,1sisk) is a partition of in with P(A) and X,-number of occurrences of 4, in independent trials, Isisk,
Random variables X, X, have a bivariate Poisson distribution if PIX, X, for any pair of nonnegative integers (k), where a,, . are nonnegative param- eters. Define a probability space and rvs X. X, on it whose joint distribution is bivariate Poisson and show that X, is a Poisson rv with mean
Prove that the random vectors X-(XX)and Y-(Y)on (FP) are independent of one another their joint df.F-F, F, and conclude that X and Y independent entails (R)(R)(R", where Vs. Fy, and are the Lebesgue Stieltjes measures determined by F. F. Fr respectively
An alternative construction of ar.v. X on a probability space with a preassigned d F is to take -[0, 1], Borel subsets of [0, 1] P-Lebesgue measure on [0, 1], and X() FL where F()-supx: F(x)
Show that foto)" [L,) for each, and (ii) over, fails. Why? 2 is Lebesgue integrable () over 0,- (0,1] for each, but that Fubini's theorem
(Young's inequality) Let be a continuous, strictly increasing function on [0,c) with 4-0. If is the inverse function (which therefore satisfies the same con ditions) then for any and equality holds iff (a) [[ 4.For j 2 1.define 4-12-17 1/2" and 1.The row sums of the double series, are simply the
Random variables (IX.21) are u if supp{\X\>\das
(Erickson) In Example 5.4.1 the function atx) = x[1-Fyldy was encountered. where F is the df of a nonnegative r.x. X. (i) Show that a(x) is nondecreasing (ii) Prove that EX- implies E(X). Hint: Ea(X) < entails alx)- of-Fix))) and hence EX/y dFiy)
Let (0.14.1-P. where pe() Lets be counting measure on the class of all subsets of (1.2). If x-e and X- prove that XX and X,X,p>1. Hint: Apply Example 1 or Example 2.1.1.
ff. are finite and nondecreasing on (-,) with / continuous, prove that ()()+d(1)(b)(+)-fm-
Let S (1.2.... I-14:45). counting measure. If X(0) 1.-counting measure on 2.IX- ses then X.0 and E X, 1 despite the fact that for 4) 1.EX. 0.Thus, Theorem 4.2.30) may fail in a e-finite measure space.
Let S (1.21, 1). (2.2.5) and Jensen's inequality fails for the convex function X
Establish that g()-(int is Riemann but not Lebesgue integrable over (-) and find a function () which is Lebesgue integrable but not Riemann integrable.
Show that the analogue of (21Xiv) for Lebesgue-Stieltjes integrals is not true in general. Construct an example for which the Riemann-Stieltjes integral over a Enite interval [a b] fails to exist.
If (X) are, Exs with a common distribution, then E max, cisalX;!= (n) H: Use Exercise 8 to establish u
If F is the df of IX, verify that for every >0 (i) Show that a sequence of rv's (X1) is uiiff sup. P[|X|1) d0 Arv. X is said to be stochastically larger than a rv. YiPLX 2x) PIY2x) for all x (ii) If the rvs X2 1, are wi and X. is stochastically larger than .. 21, then [21] is
Prove Minkowski's inequality: If X, 1,2, and p21, 1X, + X1, 5
X... and X, X for some p>0, then Xe, and EX-EXP & Iff in a finite, nondecreasing function and is a continuous nondecreasing function on [a, b], where-cache, then ---
If X, X. does X,Y XYT
Demonstrate in a confinite measure space (S. L.) that X, X does not neces sarily imply X, X. Hine: Utilize Exercise
If 5 is the set of positive integers, I is the class of all subsets of 5, and 14)-number of integers in AeX, then (SE) is a nonfinite measure space and convergence in measure is equivalent to uniform convergence everywhere.
Then Se(-1,2), whence 32 v(S) -vE), implying (5) (E). However, by 0). [0, 1) =E+)-S, so that v(S) 1, a contradic tion.
For any real x, y consider the equivalence relation xyfx-y-r-rational. Let the subset E of [0. 1) contain exactly one point of each equivalence class. Then E is a non-Lebesgue-measurable set. Hier: (i) if xe (0, 1), then xe E+ for some rin (-1,1)) (E+(E+) for distinct rationals r. s. Thus, if Eis
If (R,, ) in the Lebesgue measure space of the real line and E is a Lebesgue measurable set, so is Ex-y+xye E) and, moreover, (E+x)=(E) for every xe (-x, x)
Give an example to show that the uniqueness assertion of Theorem I is not true without the restriction of e-finiteness on. His Take (r: rrational,0 s
MX is a r.. with d.f. F. then PX-01 Pro+ P(X 0, and PIX=0 1-FO\, PLX 0 Find the df of|X|
(R,, ) is the Lebesgue measure space (of R), [0, 1], and Pl, then (P) is a probability space.
There is a 1-1 correspondence between difs and probability measures on the Borel sets of the line.
If is the Lebesgue Stieltjes measure determined by a finite nondecreasing function mon(-)and F(1) = (1-1-0 $15,prove that for finite or infinite intervals of the form (a, b),a < hand also 2.8-1-.001-00 via (la, b]] = G(b) - G(a) All-}} ==QR) G()-(-x)
Let map R-[,] into R with fir-) existing for every te R. Then g(t)= f-re R. is left continuous, ie, g) gt-rek
Prove that the extension (as defined in Theorem 1) of a measure on to the -algebra of all measurable sets is complete.
Let be a measure on a e-algebra and define [8] 0). Then is a e-algebra, where 4AN) pl4) for AA Ne (AAN: As, NC Bea, and is a complete measure on
IfXare iid.rvs with EX, &>, then ET, there is an integer n such that PiS,c) 0.Prove that if is possible and bis recurrent, then be is recurrent. Since every recurrent value is clearly possible, the set Q of recurrent values is an additive group. Show that Qis closed. Thus, if is nonempty, Q-(-, x)
() Moreover, when EX = lim Sinc iff there exists a finite [X,)-time T with ES,>- and S, X, T, = if (n 1: for 2
(Alternative proof that T., ac, implies ET.) Let (Tabe copies of T. and set T-7 Then 2-r number of times S. exceeds S, < (take S-0) and PIT. 2n+1)=PS,>$.0
(Chow-Robbins) Let (21) be positive rva with lim Y,1,ac, and ( 21 positive constants with aaa1. For e>0 define N = N,- infiesa. Prove that PIN oo, lim, ay 1) 1 and, if E sup... Y. 0) is of interest in sequential analysis (Wald). Prove that T is a stopping variable with finite moments of all orders.
Let (X) beiid. rvs with EX, > 0 and T-T-i 1:>c) for > Prove that Tea.c. If, moreover, EX c"). 0.0 <
Prove that if (X) are Lid with E X, pe(0) and N, supin 2 1:5, 5c). then Ne
Prove Corollary 3
Prove that the stopping rule 7 of (24) remains optimal when Y, X.-,c>0
Verify the second equality of (3)
Let S.-X. 21, where (X, X. EX=0,EX>0. For x>0,8>0, and 1) are Lid, random variables with 1,2... show that max 5, mas and if rs2 and
Show that lim supac. for any sequence of iid. random variables (X1) with EX>0
If T is an integrable (X)-time, where (X) are independent rvs with E X.-0, EXC
If 7, and T, are times, so are 7, T., max(7,, 7), min(T,,T), and KT,, where k is a positive integer.
Verify that if Tisan (5)-time, XF, measurable./ 1,2 1
(Heyde) Let X 211 be id %, Isp
13. (Komls-Reves) Let (X.21) be independent rvs with means E X, and positive variances satisfying lim E X, cand. Then Hi [X-EX,
(Heyde-Rogozin) I X. X. 21) are kid with (*) lim EX>> then for every sequence (h) satisfying 0 x} s (+)PX> Axl..> whence the series of Exercise 11 either converges for all C>0 or diverges for all C >
(Feller-Chung) Let P(x) be nonnegative and nonincreasing on (0) and suppose that the positive, nondecreasing sequence (b,1) satisfies (*) lim (m) as >I for some integer r or a fortiori either () 1 for some #>0 or (***) O(n). Then P(xh) either diverges for all x>0 or converges for all x>0. (Hint:
Prove that if (X, 21) are independent rxa with EX, EX2 ef, then log X, for >
9. (Klasa Teicher) If (X, X. 1) are Lid. rvs and (1) are constants with or (0-002) and hf. then (i) EXIsaia P||X|>] (1)
IS, where (X) are Lid. EXP. Hint: Recall Example Conversely, if at Hi: if whence E () () of C, then (+) ensures X e 2,(and also E X == 1) x then EXCIX for a> and E for 0 and any r.v. X, prove that X, P(X) < Hi: Employ the techniques of Theorem
Demonstrate for id. rva (X) that E sup, IX
HIX,) are iid, with EIX, for some peo), then Plim
Prove for Lid rva (X,) with S, X, that (S-C for some sequence of constants C, iff EIX,| < x
IX, are independent with E X,-a(EX)
(Chung) Let P be a positive, even function with xxxxx. If
For any sequence of rvs (S). it is always possible to find constants 0 uch that P{(5) >441
If, in the three series theorem, the alternative truncation Z, min[1,max(X-1)] is employed, then convergence of the two seriesEZ.. is equivalent to the &c. convergence of
I X. 1, are independent rvs with P{X, a 1-P(X,- -4 characterize the sequences (...) for which X, converges ac.: specialize to 10, and to a*(>0)
If (41) are independent events with PLA21 and P(A) - then P1. Hint: Ma-P(4) and X, -(1-P4, then (X1) are independent with EX, 0, EXP(4,1-1/
Thus, the restric tion of exponents in Corollary 2 is essential
Let (a) be a sequence of positive numbers with ax. Then, if p>0, there exist independent rxa X, with E X.-Q. EIX,a, wach that X, diverges ac, thereby furnishing a partial converse to Corollary 3 when 0
are id, rva, then (X) converges ac if either (i) X, is sym- metric or (ii) EX,log IX, and E X, -0.
If X.
Let (X) be air.vs. Then inf P[|X,>] > 0, for some > 0, iff inf, EX|>0
-1 if P-1.
If (X) (X) are equivalent sequences of r.vs. prove that iP X, converges if P 0,1, then PX converges) -
If X. 21, are independent rvs and Y-Xx then ac convergence of X, ensures convergence of PIX> el. EY for every >0. Conversely, convergence of these series for some e>0 guarantees ac convergence ..
Prove that if S, Isis, then for any > (24) Hint. Use the Feller-Chung lemma with 4,- (S,>). 8, (S. 5,>-E18).
For X, Y,, >1, define X, Y)-X-Y1, and show that d has all the attributes of a metric except one. For X, Ye, write X-Yi X-Yac, and let be the space of equivalence classes of . Show that is a Banach space, ea normed, linear, complete space. When p2, is a Hilbert space under the inner product (X, Y)
If (S, X. 21) are as in exercise 15 andc, then (1)max
HS-X, where (X, 1sis) are independent rvs with EX-QEX- >0, then for >> 1, Pmax 15,125PS (2) Hine: IT inflssn: $12 is,) and T-n+1 otherwise, then for 1 < < PPT - P Now apply Tchebychev's inequality.
Bernstein's Inequality. If,-X, where (X, Isis) are independent r.v.a with EX,-0. EX-6-0 which satisfy (i) EX (k/2)* for k>21 0 implies Exp (2-)),0 < 0. Now apply Exercise 4.3.3)
Hine: P)-
13. For events (4,21] and > 0, there exist distinct indices jk in [1,], where > 1/6, such that P(4,4.) p(x) - where pin) is as in Exercise
For any sequence (421) of events, define P) d-(4)-(4) Prove that (i) E Y d+ [p(n) - Pi(n)], (ii) 0 ifE Y-04-0 (ii) d.-0(*) for some >0. Hine: Utilize Exercise
Then (1/4), X, Note that ()() if E X,X,-0 Hint: Choose smallest integer 2 mm 1,2. Then (1)x,+0 asm-xand
Let (X1) bervs with (i)EX, stand (i) E(X,++XJ } = 0{n) for some a >
Prove Minkowski's inequality, that is, if X, L, XL, then IX, X, X, X Hint: Apply Holder's inequality to EIX,X, +X, when p>1.
The moment-generating function (mg) of a rv. X is the function o If (h) is finite for A->0 and -- verify that (0) Est+ (-) < < () logo) is conves in [-h] and strictly convex if X is non-degenerate (i) EX-0, then phi 21 for his ho(iv) The kth moment of X is finite for every positive integer k and
(IfXaX + Y cX+d, verify that pX, X, Y) according as ac>0 or ac
(i) Show for any, tv. X that EX (EX). (i) If P(X-1)-pc(0,1) and P(X 0 PX-2)-(1-pl/2. then, setting Z XI W-Xlus necessarily elef 6.) If X is an , tv, Y-Y(a,b) is as in Corollary 2, and Z-Yid,b), where ass& then ELY-EYP2 EZ-E2P for any p21) For a EX) E(X)+EX) (EX) and E(X) EX10 ses as in Theorem 1,
(Kochen-Stone) If (421) is a sequence of events such that for some c > 0 PA, SPA [P{4} + P{4} 0 Hint: HZ-note that E 2 SEZ,+P(4)s(1 + 4c)(EZ) for all large since EZ, co and, via Exercise 15, P4.io] Plim Z, EZ, 1) > 0
Since Y, s1+ Y neces sarily Eli-) exists. Then Theorem 2) ensures (since (Y) is u) that Elim Y, 2m E-L
15. (Kochen-Stone) If (Z. 1) is a sequence of rvs with 0 < EZ! 0 Hine: If Y and lim, (E ZEZ > ZEZ, there is a subsequence [') with E YK Replacing (a) by (e) for notational simplicity. Elim Ys K by Corollary
IfX (P) and e(X), are independent classes of events, prove that E XI, -EX-P(4), all
Let (1) be a sequence of 2,.vs, p>0 with sup[f.1X 4P: 21 and P(A) < (1) asd-0. Then X, XXX.
If the two sequences of integrable rvs (X). ) satisfy PIX, 201 XX. Y, Y. and E X, E X, finite, then E|-
IX. Xarerson(RFP) show that the indefinite integrals), X, dP J. X JP, finite uniformly for all Fiff X, converges to X in mean
Prove that X, returns to its origin, i=0 for 3
Let (X.2 11 be a simple symmetric random walk in R, that is, (X) are id random vectors such that PIX. (e) 1/2k, wheree, 0, 1, or -1 and
(i) Construct a sequence of rvs that is us from below and for which the expectation of lim X, does not exist. (u) Show that the hypothesis sup, of Corollary 3 is equivalent to X

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