Exercise 1 Let a and a' be two different consumption distributions for per- son n and...
Un=- V = Exercise 6: Let (Un) and (Vn) be two sequences such that: U. <V. aUn-1 + BVn-1 -1. 0<B<a atß. aVn-1 + BUn-1 atß 1. Let Wn = Un - Vn. Prove that Wn is a geometric sequence. Identify q and V. 2. Prove that (Un) is an increasing sequence and that (Vn) is decreasing. 3. Deduce that (Un) and (Vn) are adjacent sequences. 4. Find the limit l in terms of U, and Vo.
Exercise 17.10 Let x = V2 and for n > 1 let In+1 = 2 + In Use Banach's fixed point theorem to show that (en) converges to a root of the equation r' - 4x2 - + 4 = 0 lying between 3 and 2.
2. Let 'n ,n > l be a sequence of r.v.s such that E[Xi] μί and Var(X) σ for i-: 1, 2, , and Cov(Xi, Χ.j) Ơij for i J. Let {an ,n 1) and (bn, n 1) be the sequences of real numbers. Write down the expressions for i-l (i,Xi, Xi), Cov every i and Ơij 0 for every i j, state Var(Σί ! així), Coy(Σ, aixi, xi),
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
Exercise 6.14 Let y be distributed Bernoulli P(y = 1) unknown 0<p<1 p and P(y = 0) = 1-p f or Some (a) Show that p E( (b) Write down the natural moment estimator p of . (c) Find var (p) (d) Find the asymptotic distribution of vn (-p) as no. as n> OO.
8.2-17. Consider the distributions NĢ4x,400) and N(μ Y, 225). Let θ- obscrved mcans of two independent random sarmples, each of size n, from the respective distributions. Say we reject H0: θ 0 and accept H1: θ > 0 if x-у с. Let K(0) be the power function of the tost. ind n and c so that K(0)0.05 and K(10) 0.90, approximately. Ду-My Say x and y denote the
1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi. (a) Use CLT to get a large sample distribution of Y (b) For n = 100, give an approximation for P(Y > 100) (c) Let X be the sample mean, then approximate P(1.1 < X < 1.2) for n = 100.
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion defined by bo - 1 and bn- k-0 n E N. Show that bn-- Hint: Use a) with e*e*1 and the inverse of a power series found in the lecture.
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion...
1. Let L = {ambm cn | m <n}. Use the pumping lemma to show that L is not a CFL.
From 5.5-6. Let X1 N(4.2, 1) and X N (12,70) be two Gaussian random variables. (1) Sketch the PDFs of X1, X2 on the same chart. (2) Assuming Xi, X2 are independent, compute P[Xi > X2