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homework_7_2005

# homework_7_2005 - ε 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: ε i are iid measurement errors with zero mean and common variance σ 2 , use conditional SM analysis to find the variance of (X|Z ,…,Z ). Plot this ε 1 n conditional variance against n for σ 2 = 1 and σ 2 either 1 or 0.2. ε Useful result on the inverse of covariance matrices with a special “equicorrelated” structure. The inverse of an n × n matrix A of the type: ρ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎡ 1 1 ρ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A σ = 2 is: ] ⎡ [1 ρ + ρ − + ] [1 ] ) 2 n ( −ρ −ρ 2) -(n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 1 − A σ = 2 (1 ρ − 1 )[ + ( n − 1) ρ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1...
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