The limit state equation for a particular failure mechanism is:Z = 50+The variables have the following values:VariableX₁X₂X35X₁-4X2-2X3 + X4MeanX₁ [12.251510155s.d.The variables X1, X₂ and X3 are correlated and the covariance matrix is:X₁ X₂ X3-52X₂ -59-4X3 2 -4 6.253.53.02.51.5Determine the reliability index of the system: (i) assuming no correlationeffects; and (ii) allowing for correlation.

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Answered: The limit state equation for a…

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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Example 17-9. If T, and T2 be two unbiased estimators of y(0) with variances o,², 0,² and correlation p, what is the best unbiased linear combination of T1 and T2 and what is the variance of such a combination ?
EXAMPLE: A soft drink distributor is interested to find out if the number of cases of soft drinks ordered is related to the travel time they are delivered. The following data have been obtained from past experience. Number of Cases of Soft Drinks (X) Travel Time in Minutes (Y) 24 21 6 3 16 6 64 15 10 21 25 61 35 20 X Y 24 21 6 3 16 6 64 15 10 21 25 61 35 20 EX=180 EY=147 XY 504 ΣΧΥ = X² 576 y² 441 [Y²=
Example 2. Calculate the covariance of the following pairs of observations of the two variates X and Y. (1, 6), (2, 9), (3, 6), (4, 7), (5, 8). (6, 5), (7, 12), (8, 3), (9, 17), (10, 1). Cs Scanned with CamScanner
. 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
Determine whether A and B are independent, positively associated or negatively associated in each of the following cases : (1) N%3D 100, (А) %3 57, (B) — 65, (AB) — 32 (iї) N%3D 200, (A) %3D 108, (В) — 132, (АB) — 75 (iї) N%3D 160, (А) 3 80, (В) 3D 50, (АB) 3 25
Maximize = 3x +2x, +x, 2.x, +x, +x, s150 Subject to 2x, +2x, +8x, s 200 2x, +3x, +x, 5320 with x 20,x, 20,x, 20 4 B Beta -3 -2 -1 x3 1 3 4 5 6 Beta 1 2 3 4 5 6 B Beta 0,5 1 2 3 -3 8 1,5 -0,5 -14 250 50 -2 75 85 100 160 225 150 70 -1 200 320 170 70 x1 x2 x3 x4 x5 x6 Find the initial solution using the Method of the Minimal Element.
(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r2. (Use 3 decimal places.) r r2 x1, x2 x1, x3 x2, x3
Consider the linear model Y: = B1 + B2 d; + u;, where y; is the subjective reported happiness of individual i, d; is a dummy variable which indicates whether individual i owns a dog (d; = 1) or not (d; = 0), and u; is an unobserved error term. Assume that E(u;) = 0 and E(u;d;) = 0. We observe ani.i.d. sample of (yi, d;), i = 1,...,n, with sample size n = 100. In that sample we have no = E (1 – d;) = 80 individuals with d; = 0, and n1 reported happiness in the subpopulation with d; = 0 is no'E(1 –- d;)y; = 3, and in the subpopulation with d; = 1 is n¡' E d:y; = 8. , d; = 20 individuals with d; = 1. The average (a) Let r; = (1, d;) and 3 = (B1, B2)'. Calculate , ¤; x;, and D, '; Y;, and the OLS estimator i=1 (b) Are our assumptions above sufficient to guarantee that B, is the true causal effect of d; on y;? Explain your answer.
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 jth exam. Suppose the covariance matrix of the data is S = 52 26 52 26 52 13 26 26 65 Let y = C₁x1 + €2x2 + c3x3 be an "index" of student performance with c² + c3 + c3 = 1 and c₁ ≥ 0. Find the constants C₁, C2, C3 so that the variance of y over the data set is as large as possible.
4. The normal equations inverse matrix from a parametric adjustment is [0.15 -0.04 N |-0.04 0.26 The assumed a priori variance o = 10.0 and the a posteriori variance was calculated as ô = 100.0. Under the assumption that the scale of the a priori variance is actually unknown, what is the estimated covariance matrix of the parameters Ĉ;?

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