#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result i...
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality #2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
let n be a positive integer and let x1,...,xn be real numbers. Prove that ( x1+...+xn)2 n(x12+ x22 +...+ xn2).
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
(3) Let XXnX1,X2,⋯,Xn be iidiid random variables with Cauchy(0,1)Cauchy(0,1) distribution. That is, the density of X1 is 1/(π(1+x2)) for x∈ℜ. Prove that (X1+X2+⋯+Xn)/n is again distributed as Cauchy(0,1). The following ``answers'' have been proposed. Please read the choices very carefully and pick the most complete and accurate choice. (a) By the last exercise, the characteristic function of X1, is e−|t|e−|t|. Therefore by the fact that the Xi are iid, the characteristic function of their average is the product of n...
3. Let {x1, x2,...,xn} be a list of numbers and let ¯ x denote the average of the list. Let a and b be two constants, and for each i such that 1 ≤ i ≤ n, let yi = axi + b. Consider the new list {y1,y2,...,yn}, and let the average of this list be ¯ y. Prove a formula for ¯ y in terms of a, b, and ¯ x. 4. Let n be a positive integer. Consider...
2. a. Letỉ be the median of X1, , xn, n odd. Prove that the identity 1-1 1-1 Hold if and only if z b. Let X1, , Xn be a random sample form f(p, b), where f(p, b) is the Laplace distribution with density 1 2h2 -k-시 Assumingthat b is known and that n is odd, show that the MLE of μ is the sample median, X. (Hint: Use (a).)
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...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
2. Suppose that X1, X2, ..., Xn " N(41,01) and Yı,Y2,...,Ym * N(H2;02) are two independent random samples. (a) What is E[X - Ÿ]? (b) Find a general expression for Var[X – Ý), and use this to find an expression for the standard error ox-ý = StDev(X – Ỹ). (c) Suppose that of = 2 and o = 2.5, and also that n = 10 and m = 15. Determine the probability P(|X – Ý - (µ1 – 42)| <...
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...