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3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
Let X1, ..., Xn be a random sample from the distribution 1 f(x; 01, 02) e-(2–01)/02 x > 01, - < 01 <0, 02 > 0. 7 02 Find the method of moments estimators (MMEs) of 04 and 02.
5. Let F(x, y, z) = (yz, xz, xy) and define 2 Crin = {(x,y,z) : x2 + y2 = r2, 2 = h} Show that for any r > 0 and h ER, le F. dx = 0 Crih
Let f(x,y) = 12e-2(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFs of each?
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
Let f(x, y) 2e-(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFS of each?
Let Y be a random variable with probability density function, pdf, f(y) = 2e-2y, y > 0. Determine f (U), the pdf of U = VY.
Given the logistic map Xn+1 = run(1 – Xn) with r > 0. Show the 2-cycle is stable for 3 <r <1+V6.
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n exists a C > 0 such that for n 1,2, 3,.. Show that X is cons istent for ?