Let X and Y be two independents Exponential(1) random variables and let the random vectors U be defined as Determine Cy the correlation matrix of U. [2Y U=X L3Y]
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![Let X and Y be two independents Exponential(1) random variables and let the random vectors U be
defined as
Determine Cy the correlation matrix of U.
[2Y
U=X
L3Y]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf908fb1-8bcd-4749-bda2-35e4a7001bc6%2F1a926abc-b464-49c5-96a0-681c3f08444d%2Fe2gewt_processed.jpeg&w=3840&q=75)
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- Find the second-order contact matrix PQ mentioned in the text.By hand, compute the correlation matrix R that goes with the covariance matrix(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)
- Let X be a 4-dimensional random vector defined as X = correlation matrix E[X] = 0 0 0 0 Let Y be a 3-dimensional random vector with Rx = [X₁ X2 X3 X4]' with expected value vector and 1 0 0 Y₁ = X₁ - X2, Y₂=X2- X3, Y3 = X3 X4. -1 7 1 -1 0 0 -1 1 (a) Find a matrix A such that Y = AX. (b) Find the correlation matrix of Y, that is Ry. (c) Find the correlation matrix between X₁ and Y, that is Rx₁Y. 0 0 -1 14. 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.The mean vector and the correlation matrix of random vector X is given by 2 +4 -H. = (a) Find p12, the correlation coefficient between X₁ and X₂. E [X] E [XXT] = 5 9 1.5 -1.6 -4.8 1.5 -1.6 -4.8
- Q1: a) Consider the following functions of the random variables Y₁, Y2, Y3, and Y4 W₁ == (Y₁ + Y₂ + Y₂ +Y₂) W₂ = — (Y,₁ + Y₂ ) — — (Y₁ + Y₂ ) Find the variance-covariance matrix of W b) Set up the X matrix and ß vector for the following regression model (assume i = 1, ..., 4): Y₁ = Bo+B₁X₁1 + B₂X₁2+ B3X² il + €₁.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 A1 = [1, 0, 1], A2 = [2, 1, 1], A3 = [1, 0, 0] and the matrix A formed by these three vectors is, A = [ A1 A2 A3 ] Calculate the correlation matrix of A.
- Consider the statistical linear model Y = A0 + e with m = vations, a parameter vector 0 = (61,02, 03)" with n = 3 components and e ~ N(0; 200² I6) having mean 0 and component variance 200². A realisation of this model for data Y and design matrix A is given by 8 7 1 6 obser- €1 580.8 -140.2 2 3 1 €2 219.0 7 3 4 €3 02 Өз -64.5 6 3 5 €4 73.4 2 1 5 E5 403.8 5 7 3 €6 (a) Find the Maximum Likelihood (ML) estimate 0 of 0 and the co- variance matrix Cov(@) of the ML estimate ở (b) If 0 has prior distribution 0 ~ N(0, 10²I3), then find the distribu- tion of posterior (0|Y) using ridge regression. (c) Show that the component correlations Corr(0;|Y), (0;|Y)) of (0|Y) are smaller than those of Corr(0;, 0;) of the ML estimate 0 for i + j.For a random variable x, a new random variable y can be created by applying a linear transformation y = a + bx, where a and b are constants. If the random variable x has mean μ, and standard deviation ox, then the mean, variance and standard deviation of y are given by the following formulas. Hy = a +bμx o²y=b²o²x oy = |blox The mean annual salary for employees at a company is $35,000. At the end of the year, each employee receives a $5000 bonus and a 4% raise (based on salary). What is the new mean annual salary (including the bonus and raise) for the employees? The new mean annual salary is $.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.
