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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

=+7.10. For a given policy, let A* be the fortune of the gambler's adversary at time n.Consider these conditions on the policy: (i) W .* ≤ F+_ Fn -1; (ii) W .* ≤ A" _,; (iii)
=+7.9. Suppose that W. = 1, so that F. = F. + S ,. Suppose that p2 q and ₸ is a stopping time such that 1 ≤+ ≤n with probability 1. Show that E[F,] ≤ E[F ,, ], with equality in case p = q.
=+7.8. Here is a common martingale. Just before the nth spin of the wheel, the gambler has before him a pattern x1, ..., x, of positive numbers (k varies with n). He bets x1 + x2, or x, in case k =
=+7.7. In "progress and pinch," the wager, initially some integer, is increased by 1 after a loss and decreased by 1 after a win, the stopping rule being to quit if the next bet is 0. Show that play
=+probability of F, = F. + 1 in the fair case is 1 - 2 -*. Prove this via Theorem 7.2 and also directly.
=+7.6. In "doubling," W1 = 1, W ,, = 2W ,._ 1, and the rule is to stop after the first win.For any positive p, play is certain to terminate. Here F, - Fo + 1, but of course infinite capital is
=+7.5. Suppose that a gambler with initial fortune 1 stakes a proportion 0 (0
=+pairs. Show that this process simulates a fair coin: Z1, Z2 ,... are independent and identically distributed and P[Z; = +1] = P[Z, =- 1] =;, whatever p may be. Follow the proof of Theorem 7.1.
=+. Let D ,, be 1 or 0 according as X2 ,- 1 # X2 ,, or not, and let M, be the time of the kth 1-the smallest n such that Ej _, D, = k. Let Z, = X2M. In other words, look at successive nonoverlapping
=+(X¡(w), X2(w) ,... ) for « in C is called a collective: a subsequence chosen by any of the effective rules o contains all k-tuples in the correct proportions.
=+system o, and let C, be the w-set where every k-tuple of + 1's (k arbitrary)occurs in Y(" )(w), Y2)(c) ,... with the right asymptotic relative frequency (in the sense of Problem 6.12). Let C be the
=+An analysis of the question is a matter for mathematical logic, but one can see that there can be only countably many algorithms or finite sets of rules expressed in finite alphabets.Let Y(a), Y{")
=+there are uncountably many selection systems, how many have an effective description in the sense of an algorithm or finite set of instructions by means of which a deputy (perhaps a machine) could
=+7.3. 6.121 If V ,, is the set of n-long sequences of + 1's, the function b ,, in (7.9)maps V ,- 1 into (0, 1). A selection system is a sequence of such maps. Although
=+7.2 As shown on p. 94, there is probability 1 that the gambler either achieves his goal of c or is ruined. For p +q, deduce this directly from the strong law of large numbers. Deduce it (for all p)
=+7.1. A gambler with initial capital a plays until his fortune increases b units or he is ruined. Suppose that p > 1. The chance of success is multiplied by 1 + 8 if his initial capital is infinite
=+xx 2 ;, the overall chance of success is of course Q(x), and for an initial fortune x < ;, it is Q(2x)Q()=pQ(2x)=Q(x). The success probability is indeed Q(x) as for bold play, although the policy
=+bold-play strategy scaled down to the interval [0, ; ], and so the chance he ever reaches ; is Q(2x) for an initial fortune of x. Suppose further that if he does reach the goal of ;, or if he
=+There are policies other than the bold one that achieve the maximum success probability Q(x). Suppose that as long as the gambler's fortune x is less than 2 he bets x for x S . and 2 - x for as x <
=+-and-black is available to him. The question is not whether to gamble-he must gamble. The question is how to gamble so as to maximize the chance of survival, and bold play is the answer.
=+This example illustrates the point of Theorem 7.3. The gambler enters the casino knowing that he must by dawn convert his $100 into $20,000 or face certain death at the hands of criminals to whom
=+Example 7.9. In Example 7.2 the capital is $100 and the goal $20,000. At unit stakes the chance of successes is .005 for p = ; and 3 x 10-911 for p = . Another approximate calculation shows that
=+unit stakes or adopts bold play. At red-and-black (p = "), his chance of success with unit stakes is .00003; an approximate calculation based on (7.31)shows that under bold play his chance Q(.9) of
=+Example 7.8. The gambler of Example 7.1 has capital $900 and goal$1000. For a fair game (p =; ) his chance of success is .9 whether he bets
=+CASE 4. rs , sass. By (7.30), A(r, s) = pq + qQ (2a - 1) - pqQ(2s - 1) - pqQ(2r).From 0 ≤ 2a-1 =r +s -1 ≤ ;, follows Q(2a -1) =pQ(4a -2); and from :≤2a - = r+s- ≤ 1, follows Q(2a - 2) = p +
=+CASE 3. rsas;ss. By (7.30), A(r,s) =pQ(2a) -p[ p+qQ(2s-1)] -q[ pQ(2r)].From;< ssr+s = 2a ≤ 1, follows Q(2a) = p + qQ(4a - 1); and from 0 ≤2a - - ≤ ;, follows Q(2a -; )=pQ(4a-1). Therefore,
=+CASE 2. Sr. By the second part of (7.30), A(r,s) =qA(2r-1,2s-1)≥0.
=+CASE 1. s. By the first part of (7.30), A(r, s)=pA(2r, 2s). Since 2r and 2s have the form k/2", the induction hypothesis implies that A(2r,2s) 0.
=+For some other systems (gamblers call them "martingales"), see the problems. For most such systems there is a large chance of a small gain and a small chance of a large loss.
=+Example 7.7. Suppose as before that Fo = a and W ,, = 1, so that F ,, = a +S ., but suppose the stopping rule is to quit as soon as F ,, reaches a +b. Here F* is bounded above by a + b but is not
=+Example 7.6. The gambler has initial capital a and plays at unit stakes until his capital increases to c (0) ≤a ≤c) or he is ruined. Here Fo = a and W„ = 1, and so F =a + S, The policy is
=+6.17. Suppose that X1, X2,... are independent and P[X=0]=p. Let L,, be the length of the run of O's starting at the nth place: L,k if X,, = ... = Xn+k-1 =0X+ Show that P[L, 2, i.o.] is 0 or 1
=+6.10)Since a ,, ~ log log n (see Problem 18.17), most integers under n have something like log log n distinct prime divisors. Since log log 107 is a little less than 3, the typical integer under
=+(6.8)VI for p + q and hence that the variance of g under P ,, satisfies Var [8] 53 2 .(6.9)Psn Prove the Hardy-Ramanujan theorem:lim P. m: 8(m) - 12€ =0.
=+. 5.201 Let g(m) = E.8 (m) be the number of distinct prime divisors of m. For a = E.[ g] (see (5.46) show thata. - > 0, Show that
=+6.15. In the terminology of Example 6.5, show that log, n + log2 log2 n +@ log2 log, log2 n is an outer or inner boundary as 0 > 1 or 0 ≤ 1. Generalize.(Compare Problem 4.12.)
=+X1, X2 ,... are the successive letters produced by an information source, and h is the entropy of the source. Prove the asymptotic equipartition property: For large n there is probability exceeding
=+6.14. Shannon's theorem. Suppose that X1, X2 ,... are independent, identically dis-tributed random variables taking on the values 1 ,..., , with positive probabili-ties P .. ..., P ,. If p.(ii) = p
=+6.13. 1 A number @ in the unit interval is completely normal if, for every base b and every k and every k-tuple of base-b digits, the k-tuple appears in the base-b expansion of w with asymptotic
=+N.(u) ,..., u;) be the frequency of the k-tuple in the first n + k - 1 trials, that is, the number of I such that 1 st sn and X, = u ....., X1 + 1 =u. Show that with probability 1, all asymptotic
=+6.12. 1 Suppose that the X ,, are independent and assume the values x1, ..., x, with probabilities p(x1), ..., p(x)). For u ,,..., u. a k-tuple of the x;'s, let
=+6.11. Suppose that X1, X2, ... are m-dependent in the sense that random variables more than m apart in the sequence are independent. More precisely, let* = o(X) ...., X2), and assume that ki ,.,
=+6.10. 5.11 6.71 Suppose that (in the notation of (5.41)) B ,, - @2 - O(1 /n). Show that n"N ,, - a ,, - 0 with probability 1. What condition on B ,, - «2 will imply a weak law? Note that
=+. 1 Use the ideas of Problem 6.8 to give a new proof of Borel's normal number theorem, Theorem 1.2. The point is to return to first principles and use only negligibility and the other ideas of
=+6.8. 1 Suppose that X1, X ,,... are independent and uniformly bounded and E[ X ,, ] = 0. Using only the preceding result, the first Borel-Cantelli lemma, and Chebyshev's inequality, prove that n"'S
=+(b) Suppose that n -2S ,,; - 0 with probability 1 and that the X ,, are uniformly bounded (sup ,, | X,(w)! < w). Show that n 'S ,, - 0 with probability 1. Here the X ,, need not be identically
=+6.7. (a) Let x1, x2, ... be a sequence of real numbers, and put s ,. = x, + ""+X ,.Suppose that n "2s,2 - 0 and that the x ,, are bounded, and show that n-'s, - 0.
=+6.6. Prove Cantelli's theorem: If X1, X ,, ... are independent, E[ X ,, ] = 0, and E[ X4]is bounded, then n-'S ,, - 0 with probability 1. The X ,, need not be identically distributed.
=+. Prove Poisson's theorem: If A1, A2 ,... are independent events, p. =n"E"_P(A,), and N = E" _, /A ,, then n 'N ,, - P .. - 0.In the following problems S ,, = X; + . . . + X ...
=+6.4. For a function f on [0, 1] write ||f|| = sup, |f(x)|. Show that, if f has a continuous derivative f', then IIf - Bill sell f'll +211fll /ne2. Conclude that Il f - B,Il = 0(n-1/3).
=+let X„(w) be the number of smaller elements (between 1 and k - 1) lying to the right of k in the bottom row. The sum S ,, = X ... + . +X ,.,, is the total number of inversions-the number of pairs
=+6.3. As in Examples 5.6 and 6.3, let « be a random permutation of 1, 2 ,..., n. Each k, 1 s k & n, occupies some position in the bottom row of the permutation w;
=+. Show in Example 6.3 that P[IS, - L,| > L'/2+0.
=+6.1. Show that Z ,, - Z with probability 1 if and only if for every positive € there exists an n such that P[|Zx - Z]
=+(By the prime number theorem the ratio of the two sides in fact goes to 1.)Conclude that the rth prime p, satisfies p, xr log r and that-10(5.54)₽
=+(e) Use (5.52) and truncation arguments to prove for the number "(x) of primes not exceeding x that x(5.53)Tr (x) = log x
=+in the sense that the ratio of the two sides is bounded away from 0 and co.
=+(d) Restrict the range of summation in (5.51) to 0x
=+Use this to estimate the error removing the integral-part brackets introduces into (5.49), and show that(5.51)[ p-1 log p = log x + O(1).PSX
=+c) Show that [2n /p] - 2|n / p] is always nonnegative and equals 1 in the range n < p ≤ 2n. Deduce E2„[log*] - E„[log*] =O(1) and conclude that(5.50)[ log p =0(x).Psx
=+(b) Let log" m = E, 8 (m) log p. Show that E„[log'] =_= |2 | log p= log n +0(1).(5.49)
=+5.20. 1 (a) From Stirling's formula, deduce(5.48)E„[log] = log n +0(1).From this, the inequality E,[@ ] ≤ 2/ p, and the relation log m = Ep@ (m) log p.conclude that Epp" log p diverges and that
=+says roughly that ( p - 1)"1 is the average power of p in the factorization of large integers.
=+For a function f of positive integers, let(5.46)E.[f]=" _ f(m)m=1 be its expected value under the probability measure P .,. Show that(5.47)
=+According to (5.44), the « ,, are for large n approximately independent under P .,, and according to (5.45), the same is true of the op.
=+5.19. 2.181 For integers m and primes p, let @ (m) be the exact power of p in the prime factorization of m: m = II ,pap(m). Let 8 (m) be 1 or 0 as p divides m or not. Under each P ,, (see (2.34))
=+5.18. 2.201 The proof given for Theorem 5.3 for the special case where the u ,, are all the same can be extended to cover the general case: use Problem 2.20.
=+5.17. (a) Suppose that X, - p X and that f is continuous. Show that f(X) >p f(X).(b) Show that E[X - X,] -> 0 implies X1 , X. Show that the converse is false.
=+5.16. 1 Suppose that 0 Sp ,, ≤ 1 and put @, = min(p .,, 1 - p.). Show that, if Ea, converges, then on some discrete probability space there exist independent events A ,, satisfying P(A„) =
=+. By Theorem 5.3, for any prescribed sequence of probabilities p .,, there exists(on some space) an independent sequence of events A ,, satisfying P(A ,, ) =P ,.Show that if p ,, - + 0 but Ep ,, =
=+5.14. Let f (x) be n2x or 2n -n2x or 0 according as 0 5x
=+5.13. Let 1; = 14 be the indicators of n events having union A. Let S, = EI ,, . . . 4 ...where the summation extends over all k-tuples satisfying 1 si, < ... < in sn.Then s. = E[S ] are the terms
=+. Show that, if X has nonnegative integers as values, then E[ X] - En _, P[ X ≥ n].
=+Thus Var[n 'N.] -> 0 if and only if B ,, - @2 - 0, which holds if the A ,, are independent and P(A ,, ) = p (Bernoulli trials), because then «, = p and B ,, ==
=+(5.42)E[n \N2] =a, Var[n 'N.] - B - x2 +H-P n
=+5.11. For events A1, A2 ,..., not necessarily independent, let N1 = CX-1 /4 be the number to occur among the first n. Let 2(5.41) an=" _ P(AR), Pn = n(n-1) |sicksn E P(A,NA¡).k=1 Show that
=+5.10. 1 Minkowski's inequality is(5.40)valid for p > 1. It is enough to prove that E[(X1/P+Y1)]
=+5.9. 1 Holder's inequality is equivalent to E[ X1/Py1/4] SEV[X] E/[Y](p-1 +q == 1), where X and Y are nonnegative random variables. Derive this from (5.38).
=+(b) Show that f is convex if it has continuous second derivatives that satisfy(5.39)f1 20, f22 20, fufu >fix.
=+5.8. (a) Let f be a convex real function on a convex set C in the plane. Suppose that (X(w), Y(o)) E C for all @ and prove a two-dimensional Jensen's inequal-ity:(5.38)f(E[X], E[Y]) ≤E[f(X,Y)].
=+b) Prove that E[l/X"]≥ 1/E"[ X] for p> 0 and X a positive random vari-able.
=+5.7. (a) Write (5.37) in the form E6/"[| X|"] ≤ E[| X|")"/"] and deduce it directly from Jensen's inequality.
=+5.6. The polynomial E[({\ X| + |YD)2] in t has at most one real zero. Deduce Schwarz's inequality once more.
=+(c) By considering a random variable assuming two values, show that Cantelli's inequality is sharp.
=+b) Show that P[IX - m| 2a] ≤ 202/(a2 + a2). When is this better than Chebyshev's inequality?
=+(a) Prove Cantelli's inequality P[X-mzalso2+a2'G2@≥0.
=+5.5. Suppose that X has mean m and variance o2.
=+5.4. Suppose that X assumes the values m -a, m, m + a with probabilities p, 1 -2p, p, and show that there is equality in (5.32). Thus Chebyshev's inequality cannot be improved without special
=+5.3. Show that m = E[ X ] minimizes E[(X- m)2].
=+5.2. 2.191 Show that the unit interval can be replaced by any nonatomic probabil-ity measure space in the proof of Theorem 5.3.
=+(c) Suppose that P( A) is 0 or 1 for every A in . This holds, for example, if is the tail field of an independent sequence (Theorem 4.5), or if $ consists of the countable and cocountable sets on
=+(b) Show that, if $= {0, 2), then X is measurable & if and only if X is constant.
=+5.1. (a) Show that X is measurable with respect to the o-field { if and only if o(X) CS. Show that X is measurable o(Y) if and only if o( X) Co(Y).
=+1.1. (a) Show that a discrete probability space (see Example 2.8 for the formal definition) cannot contain an infinite sequence A1, A2 ,... of independent events each of probability ;. Since A ,,
=+(b) Suppose that 0 ≤ p ,, ≤ 1, and put «, = min{p ,,, 1 -p„}. Show that, if E ,, &, diverges, then no discrete probability space can contain independent events
=+A1, A2 ,... such that A ,, has probability p ,.
=+1.2. Show that N and Nº are dense [A15] in (0, 1].
=+1.3. 1 Define a set A to be trifling* if for each € there exists a finite sequence of intervals I ,, satisfying (1.22) and (1.23). This definition and the definition of negligibility apply as
=+(a) Show that a trifling set is negligible.
=+(b) Show that the closure of a trifling set is also trifling.

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