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Let x ER" be a Gaussian random vector with mean 0 and covariance matrix I. Prove that, for any orthogonal matrix (ie, an n × n matrix satisfying UTU-1), one has that Ur and are identically distributed.
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is an increasing sequence bounded above, that is, if for all n, xn < Xn+1 and there exists M E R such that for all n E N, Xn < M, then limny Xn = sup{Xnin EN}. (b) Prove that if Xn is a decreasing sequence bounded below, that is, if for all n, Xn+1 < xn and there exists M ER such that for...
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any a,y E R" sphere of radius 1 centered at the origin. Let x E Sn-be fixed. Let v be a random vector that is uniformly distributed in S"1. Prove: 10Vn
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
2. (24 pts) Let f(x) = >>= {* Ae Mc 1>C where A,B,C ER, A, B +0. x <C' (a) Show that f is differentiable at x = C. (b) Determine the first four terms of the Taylor series centered at x = C for f(x) using the definition of Taylor series. (c) If possible, find the Taylor series T(2) centered at x = C for f(x). (d) What's the radius and interval of convergence? (e) Find R4(C++). Can you...
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Let a_0 + (a_1)(x) + (a_2)(x^2) + ... = 1/(1 - x - x^2). Prove that the coefficients of a_n are the Fibonacci Numbers.
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
Problem 3 (7 points) Define three functions A,A,As as follows: βί(x) = 0 whenever x < 0 and A(x)-1 whenever x > 0, Moreover we let A (0)-0, β2(0)-1 and A3(0) . Let f be a bounded function on [-1, 1] (a) Prove that f ER(B1) if and only if lim0+ f(x)-f(0). In this case prove that 1 『(0) elf, -1 (b) State and prove a similar result for A. (c) Prove that f ER(B3) if and only if f...