Question: Given the random variables X1 X2 and X3 having the
Given the random variables X1, X2, and X3 having the joint density f(x1, x2, x3), show that if the regression of X3 on X1 and X2 is linear and written as
Where µi = E(Xi), σ2i = var(Xi), and σij = cov(Xi,Xj). [Proceed as on pages 386 and 387, multiplying by (x1 – µ1) and (x2 – µ2), respectively, to obtain the second and third equations.]
Answer to relevant QuestionsFind the least squares estimate of the parameter β in the regression equation µY|x = βx. Show that (a) ∑2, the random variable corresponding to 2, is not an unbiased estimator of σ2; (b) S2e = n·∑2 / n–2 is an unbiased estimator of σ2. The quantity se is often referred to as the standard error of ...Derive a (1 – α) 100% confidence interval for µY|x0, the mean of Y at x = x0, by solving the double inequality –tα/2,n–2 < t < tα/2, n–2 with t given by the formula of Exercise 14.23. In a random sample of n pairs of values of X and Y, (xi, yj) occurs fij times for i = 1, 2, . . . , r and j = 1, 2, . . . , c. Letting fi, denote the number of pairs where X takes on the value xi and fj the number of pairs ...The following are the scores that 12 students obtained on the midterm and final examinations in a course in statistics: (a) Find the equation of the least squares line that will enable us to predict a student’s final ...
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