Let x, y, a, and b elements of the reals. Suppose |x - a| < e/(2(|b| + 1), |x - a| < 1, and |y - b| < e/(2(|a| + 1) for some e > 0. Prove that |xy - ab| < e.
Let x, y, a, and b elements of the reals. Suppose |x - a| < e/(2(|b|...
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
*1. Let S2((x, y, z) e R3:xy+2 be the unit sphere and let A: S2 S, be the (antipodal) map A(x, y, z)-(-x,-y,-z). Prove that A is a diffeomorphism.
Let A C R and fA: R2-given by 1 if (x, y) E A 0 if (r, y) A Ar, y): a)Prove that fAis continuosin int(A)Uert(A) and f is dicontinuos in cl(A) b)Draw fA a) A = B2 (0) . b) A = {(x,y) | xy = 0} . c) A = {(z, y) | y E Q)
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Problem 2. Suppose that A E Fand r and y are two elements of IFn. Prove that A(x+y)-
caun cu suugroup. (2) Let (a b) be a transposition in Sn, and let a E Sp. Prove that alaba-is another trans- position. Hint: Suppose that a(z) = a, a(y) = b for some x,y € {1,...,n). Now calculate ala b)a- applied to x, y, and z where z € {1,...,n} such that a(z) a or b. (9) def o lation
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
Let X and Y have joint probability density function fX,Y (x, y) = e−(x+y) for 0 ≤ x and 0 ≤ y. Find: (a) Pr{X = Y }. (b) Pr{min(X, Y ) > 1/2}. (c) Pr{X ≤ Y }. (d) the marginal probability density function of Y . (e) E[XY].