notes_07 - Let X = X 1 where X 1 and X 2 are sub-vectors of...

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MIT OpenCourseWare http://ocw.mit.edu 1.010 Uncertainty in Engineering Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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1.010 - Brief Notes #7 Conditional Second-Moment Analysis Important result for jointly normally distributed variables X 1 and X 2 If X 1 and X 2 are jointly normally distributed with mean values m 1 and m 2 , variances σ 1 2 and σ 2 2 , and correlation coefficient ρ , then (X 1 | X 2 = x 2 ) is also normally distributed with mean and variance: σ 1 m 1|2 (x 2 ) = m 1 + ρ (x 2 m 2 ) σ 2 (1) σ 1 2 |2 (x 2 ) = σ 2 (1 −ρ 2 ) 1 Notice that the conditional variance does not depend on x 2 . The results in Eq. 1 hold strictly when X 1 and X 2 are jointly normal, but may be used in approximation for other distributions or when one knows only the first two X 1 moments of the vector X = . X 2 Extension to many observations and many predictions
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Unformatted text preview: Let X = X 1 , where X 1 and X 2 are sub-vectors of X. Suppose X has multivariate X 2 normal distribution with mean value vector and covariance matrix: m = m 1 , and Σ = Σ 11 Σ 12 ( Σ 12 = Σ 21 T ). m 2 Σ 21 Σ 22 Then, given X 2 = x 2 , the conditional vector (X 1 | X 2 = x 2 ) has jointly normal distributions with parameters: m 1|2 (x 2 ) = m 1 + Σ 12 Σ − 22 1 (x 2 − m 2 ) (2) − 1 T Σ 1|2 (x 2 ) = Σ 11 − Σ 12 Σ 22 Σ 12 Notice again that Σ 1|2 does not depend on x 2 . As for the scalar case, Eq. 2 may be used in approximation when X does not have multivariate normal distribution or when the distribution of X is not known, except for the mean vector m and covariance matrix Σ ....
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