Please do 10 & 11 Use Intermediate Value Theorem lial p(x) = x4 +7x = 9...
number 3 please Hw4.1708.pd 1 2 TL (2) LP convergence vs. convergence in probability Let Xn, nNbe a sequence of random variables and let X be another random variable. Given l < p < oo, we say that Xn converges to X in Lp if E(Xn-X") → 0 as n → x Show that this implies that Xn converges to X in probability (3) Monte Carlo Let f : 10, 1] → R be continuous and let Xn, n on...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
Use the Intermediate Value Theorem to verify that the following equation has three solutions on the interval (0,1). Use a graphing utility to find the approximate roots. 98x3 - 91x² + 25x -2=0 Let f be the function such that f(x)= 98x3 -91x2 + 25x – 2. Does the Intermediate Value Theorem verify that f(x) = 0 has a solution on the interval (0,1)? O A. No, the theorem doesn't apply because the function is not continuous. OB. Yes, the...
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the largest integer less than or equal to r. (a) Evaluate the upper and lower sums U(f, P) and L(f, P) of f with respect to or if P is the partition {0、름, î,3.3.2) of [O, 2]. 4 42 (b) Explain why f є [0,2] and use results in part (a) to give a range of fda. Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the...
2. More Rational Fun (a) Spend two to three minutes in deep meditation on Darboux's Theorem. Pay special attention to the part in bold Darboux's Theorem on Integrability Let A C R" be bounded and let f:A-R be bounded as well. Suppose E is a bounding rectangle of A. Then f is integrable over A and f-1 iff, for every ε > 0, there is a δ > 0, such that for every partition P of E with size Pll...
Question 10 RD 1 (X-μ)/μ|. Show that (5.28) 9. See Problem 5.8. Compute the signal-to-noise ratio r for the random variables from the fol. lowing distributions: (a) P(A), (b) E(n, p), (c) G(p), (d) Γ(α, β), (e) W (α, β), (f) LNue). and (g) P(α,0), where α > 2. 10. Let X and F be the sample means from two independent samples of size n from a popu- lation with finite mean μ and variance σ. Use the Central Limit...
QUESTION 6 Compute the Taylor series of f(x)= sin 2x at Then show for the series above that linck; f(x) = 0 for each r QUESTION 7 Let f (x) =-x + 3, x E [0, 1] and let P be a partition of [0,1] given by 1 2 n-1 Calculate L(P) and U(P) and prove using these summations that f is Riemann integrable on [0, 1]. Also evaluate o f(x)dx.
Let X1, . . . , Xn be independent with common density f(x) = 2x 1[0 < x < 1]. Set Vn = max(X1, . . . , Xn). (a) Verify Vn → 1 in P. (b) Show that n(1-Vn) → W in D holds for some random variable W and find the distribution function of W.