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10PDFEst

# 10PDFEst - ensemble averaging over N statistically...

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MAE 591 RANDOM DATA Probability Density Function Estimate and Errors Consider N data values { } from a transformed record x n N n , , , ,..., , = 1 2 3 x t ( ) that is stationary with x = 0 . The probability density function of x t ( ) can be estimated by ± ( ) p x N N x x = with x s 0 2 . where x is a narrow interval centered at x , is the number of data values that fall within the range of N x x x ± 2 , and the standard deviation of the sample data is given by s N x n n N = = 1 1 2 1 The bias error in the estimate is given by ± ( ) p x ε b p x x p x p x [ ± ( )] ( ) "( ) ( ) 2 24 The normalized random error in the estimate is approximated by, when it is estimated by
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Unformatted text preview: ensemble averaging over N statistically independent time history records, 1 r ˆ [ p( x )] N x p( x ) ε ≈ ∆ and, when it is computed by time-averaging a single sample record of length T with energy distributed uniformly over a bandwidth B , 1 2 r ˆ [ p( x )] BT x p( x ) ≈ ∆ Note #1: Decrease in ∆ x tends to decrease the bias error but increase the random error. Note #2: Normalized bias and random errors of are defined as ± φ [ ] ( ) ε φ φ φ b E = 1 ± − , ( ) [ ] ε φ φ φ r E E = − 1 2 1 2 ± [ ± ]...
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