(7) Let X₁, X2, X3 be jointly distributed with means #4₁ = 2, 42 = 5 and 3 = 2 andvaraince-covariance matrix4 xy260y 04herez and y are constants. Let Z₁ = X₁ + 2X₂ - 4X3 and Z₂ = X₁ + X2 + X3,determine(i) EZ,Z (Z₁, Z2)(ii) the values of z and y such that Z₁ and Z₂ are independent and Var (Z₂) = 10.Hence find the mean and variance of Z=2Z₁ +52₂-4

Answered: (7) Let X₁.X₂, X3 be jointly…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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. Let Y Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of € and assume the rows are independent. The least square estimate for 3 is 3 (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by dij. Define the sample covariance matrix of the residuals by = Prove (a) E(3) = 3 (b) cov (3,3)= Σ (3₁3) = Oik (Z'Z)-¹ 1 n-p-1
7.(0) What is the Fisher linear discriminant method. (b) Given the 2-d data below for two classes: al-[(1,1), (1,2), (1,4), (2,1), (3,1), (3,3)], m2-[(2,2), (3,2), (3,4), (5,1), (5,4), (5.5)). Determine the optimal projection line in a single dimension.
consider the following. B = {(2, -1), (-1, 1)}, B'= {(-12, 0), (-4, 4)}, [3] (a) Find the transition matrix from B to B'. [x]B' = p-1: = P = X -12 PP-1 12 (b) Find the transition matrix from B' to B. -1/12 1/4 0 0 1/4 4 (c) Verify that the two transition matrices are inverses of e 0 4 Search
6. The first derivative for Q = 15K0³[°.7 = ÔL 4.5L-03 K0.3 b. False 7. Multiplication of matrices can only occur if the no. of rows in the 1st matrix is equal to the a. True no. of columns in the 2nd matrix а. True b. False 8. In a multiple regression analysis, the we consider only one dependent variable and one independent variable only. a. True b. False 9. The following data reflect the number of times a population of business executives flew on business during the previous month: 4 9. 4 5 7 the range for these data is 5. а. True b. False 10. If p-value less than 5 % with confidence interval 95% for x variable (independent), then x variable is significance with dependent variable. a. True b. False
A researcher has estimated the following multiple regression model to investigate the determinants of capital structure in an emerging market based on data from 2016. LEV = 1.32 – 0.10TANG - 0.28PROFIT + 0.19GROWTH + e (0.92) (0.03) (0.25) (0.04) Residual sum of squares = 200Total sum of squares = 620Number of Observations = 90Standard errors of the coefficients are given in parentheses. The variables are:LEV = Leverage (total debt to total assets).TANG = Tangibility (net fixed assets to total assets). PROFIT = Profitability (net income to total assets). GROWTH = Firm growth (Percent change in sales). e = residual For each independent variable slope coefficient, test the null hypothesis that it is equal to zero against the alternative hypothesis that it is not equal to 0. The critical t value is 1.96 at the 5% significance level for a two-tailed test.
ith leverage value (hat-value) = (H)ii= hii, ith diagnal entry of hat matrix • The h¡¡ is considered to be large if hii > 2 * h = 2 * k + 1 = # of parameters, k + 1 n n = # of sample, where Ehii = k+1 h = average of diagonal entry.
Consider the following. B = {(4,-1,-4), (−3, 1, 4), (-6, 2, 10)}, B' = {(-3, -2, -24), (2, 1, 12), (4, 2, 28)}, 2 1 3 1 (a) Find the transition matrix from B to B'. [X]B' = p-1 = (b) Find the transition matrix from B' to B. P = (c) Verify that the two transition matrices are inverses of each other. E PP-1 =
Consider the following. B = {(-2, -12, -6), (1, 4, 2), (3, 12, 7)}, B' = {(−1, 1, 3), (-2, 1, 3), (-2, 1, 4)}, [x] B¹ (a) Find the transition matrix from B to B'. P-1 = P = 2 (b) Find the transition matrix from B' to B. PP-1 = -26, 9, 27 -16, 5, 14 -10, -29 (c) Verify that the two transition matrices are inverses of each other. [x] B = ↓ 1 (d) Find the coordinate matrix [x]B, given the coordinate matrix [×]B'.
Show that there is a unique choice of L and y with E = LL' + p, %3D 0.4 0.9] 0.7 with one for the population covariance matrix E = |0.4 Lo.9 0.7 %3D 1 factor m = 1
2. Using the linear model of a researcher calculated the hat-matrix = X(X'X)¯¹X' = H = TA [0.2516 0.1101 0.3931 0.1101 0.2044 0.0157 0.3931 0.1572 0.7704 0.0535 0.2421 -0.1352 0.1101 0.2044 0.0157 0.2421 0.0818 0.2232 -0.0597 0.2799 (a) Estimate o. That is, find Residual Standard Error s. /1.) zachy UNDI Yevevi TA' y = Bo + Br2 + H and the residual e using 6 observations. vull lalg 0.0818 0.0534 0.1101 0.2421 0.2044 0.2233 -0.1352 0.0157 -0.0597 0.3176 0.2421 0.2233 0.2044 0.22327 0.2233 0.2516 SOUD DAN wuc. " e = -0.02 0.12 -0.56 0.88 -0.01 [-0.07]
2. Assuming the dichotomous disease model, with Y1 and Y2 being two relatives with relatedness R, show that (a) Cov(Y1, Y2) = P(Y1 = Y2 = 1) – K². (b) express R as a function of the covariance and K.
4. Let A be a nonzero 3 x 3 matrix such that A? = 0. (i) By showing the column space of A is contained in the null space of A, show that rank(A) < nullity (A). (ii) Using your answer from part (i), show that rank(A) = 1. %3D

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