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DSC-510: Topic 2 Assignment Warm-up Activity E(X]) The notation for multivariate statistics can get quite cumbersome so it is best to be acquainted sooner rather than later. A random vector is a vector comprised of random variables X = E (X 2) X ... KA and has a mean vector denoted u = with variance/covariance matrix E = E(Xm ) var(X1) ... cov(X1, Xn)] . A multivariate random sample are multiple random vectors roped together into a matrix in a column-wise fashion and we will use matrix enumeration notation {X} } where i and j have potentially different ranges. S,, denotes the sample cov(X,, X]) ... var(X,) variance/covariance matrix and X the sample mean; it follows from univariate statistics that the unbiased estimator of E is S = - n - n - 1 ( x, - X) ( X, - X)' The correlation matrix can be found in the following matrix manipulation. Set V = diag(Vo;; ) then: R = V-isv-1 Discussion Questions 1. Use the R command X 1 - 3y2 + 13, 21 = V1 + 12 + 13 , Z2 = 4y1 + 12 -13 . Form the random vector z = (21, Z2)' . Compute each of E(x), var(x), E(z), cov(z). [20 pts] T2. Let X be drawn from a 3-dimensional normal distribution with mean vector u' = [-1, 0, 4] and variance/covariance matrix: E= -7 9 9 We want to determine what combinations or pairs of the random variables are independent. Determine if each pair of random vectors are independent: a) X1 and X2 b) X2 and X3 c) (X1, X3) and X2 Is it true that each of X1, X2, X3 are individually univariate normal given that X is multivariate normal? Give reasoning on why or why not. [40 pts]