Let l, ||2|2 = ((r)2)12 2 e ||| of folowing vectors of R. | x|10 =...
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No work, no credit, messy work, no credit, missed steps, no credit disorganized work, no credit.
Qi. Let x be a real number and u, v be the vectors u =< x,-V3x >, v =< -x,-3 > a) Find the value(s) of x if u.v 6 b) Let x v3, find the angle between the vectors u and v
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
Problem # 10, Let X be a random variable with CDF: 0 (x + 5)2/144-5 < x < 7 Ex (x) = X(r Find E(X], , and E[X"].
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].
(Proof of the Squeeze Theorem for Functional Limits). Let f.g, h: A R be three functions satisfying f(x) < 9(2) < h(r) for all re A, and suppose c is a limit point of A and lim; cf(x) = L and lim -ch() = L. Prove that lim.+c9(x) = L as well.
3. Let X1, X2, ..., Xbe iid having the common pdf S 2/r if l<r< , f(1) = 0 elswhere. Is there a real number a such that X a as n o ?
Exercise 6. Let (X, Y) be uniform on the unit ball, i.e. it has density if 2+y2 < 1, if 2y1 -{i fx.y) (r, y) Find the density of X2Y
g(x?)dx for "all" functions g: R R . Suppose that a random variable X satisfies E (g(X) = ")= ' What is P (= < x < )