Let x ER" be a Gaussian random vector with mean 0 and covariance matrix I. Prove...
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...
12. Let X be a random vector with mean y and covariance , also let A be a fixed (deterministic) matrix. Prove that E [X'AX] = tr(AE) + M'Aj
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 -
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...
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y. (c) Find CY, the covariance matrix of Y d) Find P(Y 2). 5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y....
Q2) All sub problems are related. Show all steps for full credit. Let U and V be independent and identically distributed (i.i.d.) Gaussian(0,2) (mean = 0, and standard deviation 2) random variables. The (2x1) random vector X is given as X = II a) Find the covariance matrix of the random vector X. (10 points) . Find the expected value b) A (2x1) derived random vector Y = 2 is given as Y = AX where A = [1 vector...
3. Let N = (M, ,X,) be a multinomial (mi pı, pr) random vector. Compute the PT mean and covariance matrix of a N. That is, find E(N) and COV (N, N) for i,j- 1,... ,r. Computing the latter can be done directly (least recommended), by expressing N, as an appropriate sum of Bernoulli RVs, or by looking at N N, 3. Let N = (M, ,X,) be a multinomial (mi pı, pr) random vector. Compute the PT mean and...
(12 points) The random variables X1, X2, and X; are jointly Gaussian with the following mean vector and covariance matrix: 54 2 07 2 5 -1 0-1 The random variable Y is formed from X1, X2, and X; as follows: Y=X1 - X2 + X3 +4. Determine P( Y> 3).
5. Determine the characteristic function Ø(u) = E(exp(ju? x)] of the Gaussian random vector X having mean m= [1 3]T and covariance matrix C = [? 2]