Two random variables Z and W are defined as Z = X + aY and W = X - aY, where X and Y are also random variables, Find 'a' such that Z and W are orthogonal.
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- Let X = (X₁, . . . , Xn) denote an (n × 1) vector of independent random variables Xi∼ N (0, 1), i = 1, . . . , n. Let A = [aij] denote an (m × n) matrix, and define a random vector Y = (Y₁, . .. , Ym) asY = AX. Show that Cov(Yi, Yj) Which of the following lines is a valid argument? See attached picLet X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A is a fixed m by n matrix and b is a fixed m-dimensional vector. Show that CY = ACXAT .The joint pmf of X and Y is f(x, y) = 1/6, 0 ≤ x+y ≤ 2, where x and y are nonnegative integers. Compute Cov(X, Y) and determine the correlation coefficient.
- If supX∈S ||X||p 1, then S is uniformly integrable.Recall that we say that a random variable X is in the vector space L^2 if it has finite second moment, EX^2 In this problem we will understand a bit better the geometry of the vector space L^2(1) Show that ||X||_2 = √EX^2 is a normIs it true that if x is orthogonal to v and w, then x is also orthogonal to v − w?
- Show {s,t,4s-3t)|s and t are in ℝ} is a subspace of ℝ3.Let hat(\beta ) be the (k+1)\times 1 vector of OLS estimates.(i) Show that for any (k+1)\times 1 vector b, we can write the sum of squared residuals asSSR(b)=hat(u)^(')hat(u)+(hat(\beta )-b)^(')x^(')x(hat(\beta )-b).{():} Hint: Write (y-xb)^(')(y-xb)=[hat(u)+x(hat(\beta )-b)]^(')[hat(u)+x(hat(\beta )-b)] and use the fact that{:x^(')u=0.}6 Consider the setup of the Frisch-Waugh Theorem.(i) Using partitioned matrices, show that the first order conditions (x^(')x)hat(\beta )=x^(')y can be written asx_(1)^(')x_(1)hat(\beta )_(1)+x_(1)^(')x_(2)hat(\beta )_(2)=x_(1)^(')yx_(2)^(')x_(1)hat(\beta )_(1)+x_(2)^(')x_(2)hat(\beta )_(2)=x_(2)^(')y.(ii) Multiply the first set of equations by x_(2)^(')x_(1)(x_(1)^(')x_(1))^(-1) and subtract the result from the second setof equations to show that(x_(2)^(')M_(1)x_(2))hat(\beta )_(2)=x_(2)^(')M_(1)y,where M_(1)=I_(n)-x_(1)(x_(1)^(')x_(1))^(-1)x_(1)^('). Conclude thathat(\beta )_(2)=(x_(2)^(¨)^(')x_(2)^(¨))^(-1)x_(2)^(¨)^(')y.Random vector X has PDF fX(x) = {ca'x when 0≤x≤1 and 0 otherwise} where a=[a1,...,an]' is a vector with each component ai > 0. What is c?
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