Let X be a Gaussian random vector with distribution [1 0 0 (86) 1 0 0 1 Let Ý be a Gaussian random vector formed by multiplying X by a certain matrix: M=61 X~N Find the covariance matrix of Y. 1 X₂
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- Find the second-order contact matrix PQ mentioned in the text.4. Let X'= (X₁,..., Xn) be an n-dimensional random vector whose covariance matrix exists. Let A be an m x n matrix of constants. Then Cov(AX) = ACov(X)A'. True or false, give reasoning.By hand, compute the correlation matrix R that goes with the covariance matrix
- Suppose three tests are administered to a random sample of college students. Let X1, ..., XN be observation vectors in R3 that list the three scores of each student, and for j = 1, 2, 3, let xj denote a student's score on the j th exam. Suppose the covariance matrix of the data is 2 6. Let y be an “index" of student performance, with y = c,x1 + C2X2 + C3X3 and cỉ + c² + c = 1. Choose c1, C2, C3 %3D %3D so that the variance of y over the data set is as large as possible. [Hint: The eigenvalues of the sample covariance matrix are 1 = 3, 6, and 9.1Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.4. Let X1, X2, X3, X4 denote the variables bill length, bill depth, flipper length and body mass, respectively. Suppose that we introduce two new variables: Y1 = 3X1 + 2X2 and Y2 = X2 + X3 + X4. Let Y be the dataset recording variables Y1 and Y2 of the same individuals as in X. Find the mean vector and covariance matrix of Y.
- Explain Homoskedasticity-Only Standard Errors of the covariance matrix?Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.2 4 -4 -2 V3 = 6 O Here are three vectors: vị = V2 = 3 3 -2
- If X and Y are independent random variables with means and variances (x, σ²) and (µ¸, σ3), respectively, then what are the mean and variance of Z = aX + bY? For vectors x and y and a tall matrix H where y = Hx, what is the least squares estimate of x given the data y.Use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector.Estimation of the model resulted in the following least squares estimates and corresponding estimated covariance matrix by var (b)) cou (b, bz, b3) = cou (b, ba) Cou (bị , ba) cov (bị , b3 ) Cou (b, , b3) var (b3) 1 var (b2) COu (b) , b3) cov (b2 , b3) 1 4. Which of the following is correct? O The hypothesis Họ : B2 = 0 versus the alternative H¡ : B2 ± 0 is not rejected using a 5% significance level and the hypothesis Ho : B2 = 1 versus the alternative H1 : Bz 1 is rejected using a 5% significance level O The hypothesis Họ : Bz = 0 versus the alternativeH : 32 ± 0 is rejected using a 5% significance level and the hypothesis Ho : B2 =1 versus the alternative H¡ : B2 # 1 is not rejected using a 5% significance level O The hypothesis Họ : B2 = 0 versus the alternative Hj : 32 ± 0 is not rejected using a 5% significance level and the hypothesis Ho : B2 = 1 versus the alternative H1 : B2 # 1 is not rejected using a 5% significance level %3D None of the above The hypothesis Họ : B2 = 0…