Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance ...
Q3. Assume that X- (X1, X2) is multivariate normal with mean zero and the variance-covariance matrix Let λ-(A1, λ2), Ai, A2 Value-at-Risk for Y 0, Ai + λ2-1. Let Y-Aix, + λ2Xy. Find the weights λι, λ2 that minimize
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....
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 ....
Let (?,?) have a bivariate normal distribution with mean (0,0) and covariance matrix . Let (?1,?1),…,(??,??) be a random sample of size n from this distribution. Find a sufficient statistic for p.
x={x1,x2,x3} has the 3-variate normal dustribution with mean 0 and variance covariance matrix=(3 1 1 1 3 1 1 1 4) find PDF of x in full
Thank you Assume that Y is a 3 × 1 random vector with mean vector ,y = μ and covariance matrix ΣΥΥ-σ2 . I. Assume that e is an independent random variable variable with zero mean and variance ф2 . Derive the mean and variance for W-2 1 Y + 5. Derive the covariance matrix between W and Y 6. Derive the correlation matrix between Wand Y. 7. Derive the variance covariance matrix for V- W Y, i.e., derive
X1 Let X = | ' | | be a normal random vector with the following mean vector and covariance matrix 2J 1 2 Let also 1 2 a. Find P(O X21) b. Find the expected value vector of Y, mys Y
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...
1.4 Suppose x is a random vector drawn from a d-dimensional multivariate Gassian distribution with mean 0 and covariance Σ Define y := Qx + u, for a known (invertible) d × d matrix Q, and a dx 1 vector v. What is the distribution of y?
MULTIVARIATE DISTRIBUTIONS 3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)