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Question 25 2 pts Let C be the path from (0,0) to (1, 0) to (1,2)....
8. Let X = {fe (C[0, 1], || ||00): f() = 1} and Y = {fe (C[0, 1], || |co) : 0 <f() < 1}. Show that X is complete but Y is not complete .
Question 26 1 pts Use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the given integral. Let F = < yz, xz, xy). Find the work done by this force field on an object moving from (1,1,1) to (4,4,4). O 54 57 60 o oo 63
4. [10 pts] Let X be a random variable with probability density function if 1 < a < 2, 2 f(a)a 0 otherwise. Find E(log X). Note: Throughout this course, log = loge.
Given the path C: x(t) = (cost, sint, t), 0<t<2n. Let f(t, y, z) = x2 + y2 + 22. Evaluate (12 pts) f(,y,z)ds.
5. Let X1,...,Xn be a random sample from the pdf f(\) = 6x-2 where 0 <O<< 0. (a) Find the MLE of e. You need to justify it is a local maximum. (b) Find the method of moments estimator of 0.
Question 5 (1 point) S2x4, Let f(2) - <x< 0 5 sin(x), 0 < x < Evaluate the definite integral [ f(x) f(x)dx. 5 O + 10 873 - 10 O 1/25 - 10
let a,b > 0 . Prove that
DI < Val
7. Let V = P2-{polynomials in x of degree 2 on the interval o <エく1) and let H span(1,2}, Find the vector in H (i.e., the linear function) that is closest to a2 in the sense of the distance
Question 3 1 pts Let 7 = (xy, - xy) and let D be given by 0 < x <1, 0<y<1. Compute Sap Ē. dr. 0-1 OO O1 O2
(1) Let {fn} < C[a, b], and let {xn} c [a, b]. Suppose that fn + f uniformly on [a, b] and In + x (as n +00. Show that limn7 fn(2n) = f(x). [3] To a + + ( f or intuico te fan a ant [Dannt u