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Questions and Answers of Theory Of Probability

=+17. Apply Proposition 15.5.3 and prove that A[ϕ(N)/N] = ζ(2)−1
=+16. In Proposition 15.5.3, show that the assumption that f(n) is nonnegative can be relaxed to infn f(n) > −∞ or supn f(n) < ∞.
=+15. Derive formula (15.9).
=+14. If limn→∞ f(n) =c, then demonstrate that A(f) = lims→1 Es[f(N)] = c.
=+13. Suppose q is a positive integer and cm is a sequence of numbers with cm+q = cm for all m. Prove that the series n m=1 cm m converges if and only if q m=1 cm = 0. (Hint: Apply Proposition
=+12. Liouville’s arithmetic function is defined byλ(n) = * 1, n = 1(−1)e1+···+ek , n = pe1 1 ··· pek k .Prove thatd|nλ(d) = %1 if n is a square 0 otherwise.
=+11. Demonstrate that neither ϕ(n) nor μ(n) is completely multiplicative.Show that the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative.
=+10. Verify that the identities 1 = d|nμ(d)τ(n/d)n = d|nμ(d)σ(n/d)hold for all natural numbers n.
=+9. Check that the Dirichlet inverse f[−1] of a multiplicative arithmetic function f is multiplicative. (Hints: Assume otherwise, and consider the least product mn of two relatively prime natural
=+n=1 ln n ns ≥ 2∞n=1 ln(2n)(2n)s − ln 2 2s∞n=1 ln n ns ≤ 2∞n=1 ln(2n)(2n)s .Note that the second inequality in (15.11) can fail when n = 1 and s < 1/ ln 2. In this special case, apply
=+ln(2n)(2n)s ≤ ln(2n − 1)(2n − 1)s + ln(2n)(2n)s 2ln(2n)(2n)s ≥ ln(2n + 1)(2n + 1)s + ln(2n)(2n)s (15.11)for appropriate values of n and s. Use these to prove that∞
=+8. Prove the two inequalities (15.6). (Hints: Show that the function f(t) = t−s ln t is decreasing on the interval [e1/s, ∞). Use this to prove that 2
=+7. Suppose the independent realizations M and N of Zipf’s distribution generate arithmetic functions Y = f(M) and Z = g(N) with finite 392 15. Number Theory expectations. Show that the random
=+What happens in the exceptional case when all ni = 1?
=+6. Let N1,...,Nm be an i.i.d. sample from the Zipf distribution with values n1,...,nm. If at least one ni > 1, then prove that the maximum likelihood estimate of s is uniquely determined by the
=+5. Suppose the arithmetic function Y = f(N) satisfies Es(Y ) = 0 for all s ≥ r > 1. Show that Y ≡ 0. (Hint: Prove that f(n) = 0 for all n by induction and sending s to ∞ in the equation ns
=+4. Choose two independent random numbers M and N according to Zipf’s distribution. Prove that M and N are relatively prime with probability ζ(2s)−1. (Hints: Let Xp and Yp be the powers of the
=+3. Let N follow Zipf’s distribution. Demonstrate that Es(Nk) = ζ(s − k)ζ(s)Es(Nk | q divides N) = qkζ(s − k)ζ(s)Es(Nk | N prime) =p pk−sp p−s for k a natural number with s>k + 1.
=+2. Show that the gap g between successive prime numbers can be arbitrarily large. (Hint: Consider the sequence of numbers beginning with(g + 1)! + 2.)
=+1. Demonstrate that∞n=21 − 1 n2= 1 2∞n=01 + z2n= 1 1 − z , |z| < 1.
=+14. Verify formula (14.13) by induction on s
=+13. In the coupling method demonstrate the bound Pr(S > 0) ≥ α∈I pα1 + E(Vα).See reference [170] for some numerical examples. (Hints: Choose T appropriately in equality (14.8) and apply
=+β∈Nα pαpβ ≤ λ2(2t + 1)/n + 2λpt. (Hint:b1 = p2t + 2tp2t(1 − p)+ [2nt − t 2 + n − 3t − 1]p2t(1 − p)2 exactly. Note that the pairs α and β entering into the double sum for b1
=+β∈Nα\{α} pαβ = 0. Finally, show that b1 = α∈I
=+t−1 k=j Wk for j > 1. The number of such success runs starting in the first n positions is given by S = α∈I Xα, where the index set I = {1,...,n}.The Poisson heuristic suggests the S is
=+12. Consider an infinite sequence W1, W2,... of independent, Bernoulli random variables with common success probability p. Let Xα be the indicator of the event that a success run of length t or
=+11. In the somatic cell hybrid model, suppose that one knows a priori that the number of assay errors does not exceed some positive integer d.Prove that assay error can be detected if the minimum
=+14.5 Problems 371 regardless of which β ∈ Nα\{α} is chosen [76]. Setting r = p(1 − p), verify the recurrence relation wn+1,d12,d13 = r(wn,d12−1,d13 + wn,d12,d13−1 + wn,d12−1,d13−1)+
=+10. In the somatic cell hybrid model, suppose that the retention probability p = 1 2 . Define wn,d12,d13 = Pr[ρ(Cn 1 , Cn 2 ) = d12, ρ(Cn 1 , Cn 3 ) = d13]for a random panel with n clones. Show
=+Now define the neighborhoods Nα so that Xα is independent of those Xβ with β outside Nα. Demonstrate thatα∈Iβ∈Nαpαpβ =n d n d−n − d d 1
=+ This is a special case of the Poisson approximation treated in Example 14.2.2 by the coupling method. In this exercise we attack the birthday problem by the neighborhood method. To get started,
=+9. Suppose n balls (people) are uniformly and independently distributed into m boxes (days of the year). The birthday problem involves finding the approximate distribution of the number of boxes
=+8. A graph with n nodes is created by randomly connecting some pairs of nodes by edges. If the connection probability per pair is p, then all pairs from a triple of nodes are connected with
=+370 14. Poisson Approximation(Hints: Let I be the set of all 2n vertices, Xα the indicator that vertexα has all of its edges directed toward α, and Nα = {β : β −α ≤ 1}.Note that Xα is
=+7. Consider the n-dimensional unit cube [0, 1]n. Suppose that each of its n2n−1 edges is independently assigned one of two equally likely orientations. Let S be the number of vertices at which
=+6. In the context of Example 14.3.1 on the law of rare events, prove the less stringent boundπS − πZTV ≤ nα=1 p2αby invoking Problems 29 and 30 of Chapter 7.
=+α. Calculate an explicit Chen-Stein bound, and give conditions under which the Poisson approximation to S will be good.
=+5. In certain situations the hypergeometric distribution can be approximated by a Poisson distribution. Suppose that w white balls and b black balls occupy a box. If you extract n
=+4. In the m´enage problem, prove that Var(S)=2 − 2/(n − 1).
=+3. In Problem 2 prove that the total variation inequality can be improved toπS − πZTV ≤2n+1 + e−1(n + 1)! .This obviously represents much faster convergence. (Hints: Use the exact
=+2. For a random permutation σ1,...,σn of {1,...,n}, let Xα = 1{σα=α}be the indicator of a match at position α. Show that the total number of matches S = nα=1 Xα satisfies the coupling
=+1. Verify that λ−1(1 − e−λ) ≤ min{1, λ−1} for all λ > 0.
=+22. Prove that the linear density fij0 + fij1x is nonnegative throughout the interval (aij , ai,j+1) if and only if its center of mass cij lies in the middle third of the interval. (Hint: Without
=+21. Consider a continuous-time branching process in which the possible number of daughter particles is bounded above a common integer b for all particle types. Show how the process can be
=+354 13. Numerical Methods count process Xt moves from state to state by randomly selecting two genes, which may coincide. The first gene dies, and the second gene reproduces a replacement. If the
=+20. In Moran’s population genetics model, n genes evolve by substitution and mutation. Suppose each gene can be classified as one of d alleles, and let Xti denote the number of alleles of type i
=+19. In counting jumps in a Markov chain, it is possible to explicitly calculate the matrix exponential (13.12) for a two-state chain. Show that the eigenvalues of the matrix diag(Λ) + C(u)
=+18. A square matrix M = (mij ) is said to be diagonally dominant if it satisfies |mii| > j=i |mij | for all i. Demonstrate that a diagonally dominant matrix is invertible. (Hint: Suppose Mx = 0.
=+(e) Inverse transform to find the solutions pj (t).(f) Compute ˆdk, and show that ˆd0 = 0 and all other ˆdk are negative.(g) Deduce that limt→∞ pj(t)=ˆp0(0) = 1/n for all j.
=+a ∗ bj into the pointwise product naˆkˆbk.(d) Solve the transformed equations ˆp k(t) = npˆk(t) ˆdk.
=+(c) Prove that the finite Fourier transform maps the convolution
=+13.8 Problems 353 17. Consider n equally spaced points on the boundary of a circle. Turing suggested a simple model for the diffusion of a morphogen, a chemical important in development, that
=+16. In the coin tossing example, prove that the probabilities un satisfy un = qpr 1 − pr + O(r−n)for any r
=+15. Let F(s) = ∞n=1 fnsn be a probability generating function. Show that the equation F(s) = 1 has only the solution s = 1 on |s| = 1 if and only if the set {n: fn > 0} has greatest common
=+14. For a complex number c with |c| > 1, show that the periodic function f(x)=(c − e2πix)−1 has the simple Fourier series coefficients ck = c−k−11{k≥0}. Argue from equation (13.5) that
=+13. Continuing Problems 11 and 12, suppose a constant a ≥ 0 and positive integer p exist such that|ck| ≤ a|k|p+1 for all k = 0. Integration by parts shows that this criterion holds if
=+352 13. Numerical Methods for |k| < n. Functions analytic around 0 automatically possess Fourier coefficients satisfying the bound |ck| ≤ ar|k|.
=+12. Continuing Problem 11, let ck be the kth Fourier series coefficient of a general periodic function f(x). If |ck| ≤ ar|k| for constants a ≥ 0 and 0 ≤ r < 1, then verify using equation
=+11. For 0 ≤ m ≤ n − 1 and a periodic function f(x) on [0,1], define the sequence bm = f(m/n). If ˆbk is the finite Fourier transform of the sequence bm, then we can approximate f(x) by
=+10. From a periodic sequence ck with period n, form the circulant matrix C =⎛⎜⎜⎝c0 cn−1 cn−2 ··· c1 c1 c0 cn−1 ··· c2... ... ... ...cn−1 cn−2 cn−3 ··· c0⎞⎟⎟⎠
=+9. Consider a power series f(x) = ∞m=0 cmxm with radius of convergence r > 0. Prove that∞m=k mod n cmxm = 1 nn−1 j=0 u−jk n f(uj nx)for un = e2πi/n and any x with |x| < r. As a special
=+8. Prove parts (a) through (c) of Proposition A.3.1.
=+7. For 0 ≤ r < n/2, define the rectangular and triangular smoothing sequences cj = 1 2r + 1 1{−r≤j≤r}dj = 1 r1{−r≤j≤r}1 − |j|rand extend them to have period n. Show that cˆk = 1
=+6. Show that the sequence cj = j on {0, 1,...,n − 1} has finite Fourier transform cˆk =* n−1 2 k = 0−1 2 + i 2 cot kπn k = 0.
=+5. Explicitly calculate the finite Fourier transforms of the four sequences cj = 1, cj = 1{0}, cj = (−1)j, and cj = 1{0,1,...,n/2−1} defined on{0, 1,...,n − 1}. For the last two sequences
=+4. Assume X and Y are independent random variables whose ranges are the nonnegative integers. Specify a finite Fourier transform method for computing the discrete density of the difference X −Y .
=+3. Suppose you are given a transition probability matrix P and desire the n-step transition probability matrix P n for a large value of n.Devise a method of computing P n based on the binary
=+2. Show that the entries of the block Gauss-Seidel update (13.3) are nonnegative.
=+1. Prove that the matrix R defined by equation (13.1) is invertible.(Hints: Apply Proposition 7.6.1 and the Sherman-Morrison formula(A + uvT )−1 = A−1 − A−1uvT A−1 1 + vT A−1u for the
8.11 Show that (B1) implies (a) (8.24) and (b) (8.26).
8.10 (a) If sup |Yn(t)| P→ 0 and sup |Xn(t) − c| P→ 0 as n → ∞, then sup |Xn(t) −ceYn(t)| P→ 0, where the sup is taken over a common set t ∈ T .(b) Use (a) to show that (8.22) and
8.9 Prove Lemma 8.7.
8.8 Let X1,...,Xn be iid as N(θ , 1) and consider the improper density π(θ) = eθ4. Then, the posterior will be improper for all n.
8.7 Prove the result stated preceding Example 8.6.
8.6 Give an example in which the posterior density is proper (with probability 1) after two observations but not after one.[Hint: In the preceding example, let π(τ )=1/τ 2.]
8.5 Let X1,...,Xn be independent, positive variables, each with density (1/τ )f (xi/τ ), and let τ have the improper density π(τ )=1/τ (τ > 0). The posterior density after one observation is a
8.4 In Example 8.5, the posterior density of θ after one observation is f (x1 − θ); it is a proper density, and it satisfies (B5) provided Eθ |X1| < ∞.
8.3 The assumptions of Theorem 2.6 imply (8.1) and (8.2).
8.2 Referring to Example 8.1, consider, instead, the minimax estimator δn of p given by(1.11) which corresponds to the sequence of beta priors with a = b = √n/2. Then,√n[δn − p] = √nX n
8.1 Determine the limit distribution of the Bayes estimator corresponding to squared error loss, and verify that it is asymptotically efficient, in each of the following cases:(a) The observations
7.34 The derivatives of all orders of the density (7.37) tend to zero as x → ξ .
7.33 Let X1,...,Xn be iid according to the three-parameter lognormal distribution(7.37). Show that
7.32 In Example 7.15, (a) verify equation (7.39), (b) show that the choice a = −2 produces the estimator with the best second-order efficiency, (c) show that the limiting risk ratio of the MLE (a =
7.31 In the preceding problem, compare (a) the asymptotic distribution of the MLE and the UMVU estimator of c; (b) the normalized expected squared error of these two estimators.
7.30 In Example 7.13, determine the UMVU estimators of a andc, and the asymptotic distributions of these estimators.
7.29 In Example 7.13, show that(a) cˆ and aˆ are independent and have the stated distributions;(b) X(1) and log[Xi/X(1)] are complete sufficient statistics on the basis of a sample from (7.33).
7.28 In Example 7.12, show that (a) √n(ˆbˆ −b) L→ N(0, b2) and (b) √n(bˆ −b) L→N(0, b2).
7.27 In Example 7.12, compare (a) the asymptotic distributions of ξˆ and δn; (b) the normalized expected squared error of ξˆ and δn.
7.26 For each of the following estimates, write out the ψ function that determines it, and show that the estimator is consistent and asymptotically normal under the conditions of Theorems 9.2 and
7.25 Theorem 9.3 Under the conditions of Theorem 9.2, if, in addition(i) Eθ0∂∂t ψ(X, t)|t=t0!is finite and nonzero,(ii) Eθ0ψ2(X, t0)!< ∞, then√n(T0 − t0) L→ N(0, σ2 T0), where σ2 T0
7.24 To have consistency of M-estimators, a sufficient condition is that the root of the estimating function be unique and isolated. Establish the following theorem.Theorem 9.2 Assume that conditions
7.23 Let F have a differentiable density f and let ψ2f < ∞.(a) Use integration by parts to write the denominator of (7.27) as [ ψ(x)f (x)dx]2.(b) Show that σ2(F,ψ) ≥ [(f /f )2f ]−1 = I
7.22 Show that if ρ s defined by (7.24), then ρ and ρ are everywhere continuous.
7.21 Show that the likelihood (7.21) is unbounded.
7.20 Verify the roots (7.22).
7.19 Suppose that in (7.21), the ξ ’s are themselves random variables, which are iid as N(µ, γ 2).(a) Show that the joint density of the (Xi, Yi) is that of a sample from a bivariate normal
7.18 When τ = σ in (7.21), show that the MLE exists and is consistent.
7.17 In Example 7.8:(a) Show that for j > 1 the expected value of the conditional information (given Xj−1)that Xj contains about β is 1/(1 − β2).(b) Determine the information X1 contains about
7.16 (a) In Example 7.8, show that the likelihood equation has a unique solution, that it is the MLE, and that it has the same asymptotic distribution as δn = n i=1 XiXi+1/ n i=1 X2 i .(b) Show
7.15 Prove that the sequence X1, X2,... of Example 7.8 is stationary provided it satisfies(7.17).

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