1.4 Suppose x is a random vector drawn from a d-dimensional multivariate Gassian distribution with mean...
Suppose X is a random vector, where X = (X(1), . . . , x(d))T , d with mean 0 and covariance matrix vv1 , for some vector v ER 1point possible (graded) Let v = . (i.e., v is the normalized version of v). What is the variance of v X? (If applicable, enter trans(v) for the transpose v of v, and normv) for the norm |vll of a vector v.) Var (V STANDARD NOTATION SubmitYou have used 0...
Suppose that x is a p-dimensional random vector with mean y and covariance ? = UDUt where U = Ui U2 ... Up li dz ... D = de with u1,...,Up orthonormal. Show that S di Cov(u: (x – u), uf(x – u)) = { 0 i= otherwise
Consider a random vector X e RP with mean EX is a p x p dimensional matrix. Denote the jth eigenvalue and jth eigenvector of as and øj, respectively. 0 and variance-covariance matrix Cov[X] = . Note that Define the random score vector Z as Х,Ф — Z where is the rotation matrix with its columns being the eigenvectors 0j, i.e., | 2|| Ф- Perform the following task: Show that the variance-covariance matrix of random score vector Z is ....
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
The random vector x (XI, X2,... ,Xk)' is said to have a symmetric multivariate normal distribution if x ~ Ne(μ, Σ) where μ 1k, i.e., the mean of each X, is equal to the same constant μ, and Σ is the equicorre- lation dispersion matrix, i.e. when k 3, μ-0, σ2-2 and ρ 1/2, find the probability that Hint: Recall that if x = (Xi, , Xk), has a continuous symmetric dis tribution, then all possible permutations of X1,... ,Xk...
2. Suppose that you can draw independent samples (U,, U2,U. from uniform distribution on [0,1]. (a) Suggest a method to generate a standard normal random variable using (U, U2,Us...) Justify your answer. b) How can you generate a bivariate standard normal random variable? (Note that a bivariate standard normal distribution is a 2-dimensional normal with zero mean and identity covariance matrix.) (c) What can you suggest if you want to generate correlated normal random variables with covariance matrix Σ= of...
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1 3 m < n, let X1 denote the vector consisting of the last n - m coordinates of X < n, let 1 (a). Find the mean vector and the variance-covariance matrix of X1. (b). Show that Xi is a (n- m)-dimensional normal random vector.
Let V be a m-dimensional Gaussian random vector with zero-mcan and covariance In. Let X = ЛⅤ and Y = CV. Find E(X | Y). TL - Let V be a m-dimensional Gaussian random vector with zero-mcan and covariance In. Let X = ЛⅤ and Y = CV. Find E(X | Y). TL -
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...