homework_7_2005

homework_7_2005 - i are iid measurement errors with zero...

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Homework Set #7 Problem 1 Consider a sequence of random variables X 1 , X 2 , …, X i , …, for example denoting the monthly profits of a supermarket chain. Suppose that X i ~ (m, σ 2 ) for all i and that the correlation coefficient between X i and X j , ρ ij , depends only on the time lag |i-j| as j| ρ ij = 0.8 |i- Using conditional SM analysis, calculate and plot, as a function of k 1, the variances of (X i+k |X i ) and (X i+k |X i ,X i-1 ). Comment on the results. Problem 2 X is an unknown quantity, say the compressive strength of a concrete column, with mean value m and variance σ 2 . Several indirect measurements of X, in the form Z i = X + ε i for i = 1, …, n, are made through a nondestructive technique. Under the assumption that the
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Unformatted text preview: i are iid measurement errors with zero mean and common variance 2 , use conditional SM analysis to find the variance of (X|Z ,,Z ). Plot this 1 n conditional variance against n for 2 = 1 and 2 either 1 or 0.2. Useful result on the inverse of covariance matrices with a special equicorrelated structure. The inverse of an n n matrix A of the type: 1 1 A = 2 is: ] [1 + + ] [1 ] ) 2 n ( 2) -(n 1 A = 2 (1 1 )[ + ( n 1) 1...
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This note was uploaded on 12/04/2011 for the course ESD 1.151 taught by Professor Danieleveneziano during the Spring '05 term at MIT.

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