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briefnts7_con_sm

briefnts7_con_sm - ⎣ X 2 ⎦ normal distribution with...

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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: m 1|2 (x ) = m ρ + σ 1 (x m 2 ) 2 1 σ 2 2 (1) 2 2 σ 1|2 (x ) σ = (1 ρ 2 ) 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 X 1 Let X = , where X 1 and X 2 are sub-vectors of X. Suppose X has multivariate
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Unformatted text preview: ⎣ X 2 ⎦ normal distribution with mean value vector and covariance matrix: ⎡ m 1 ⎤ m = ⎢ ⎥ , 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: − 1 ⎧ m 1|2 (x 2 ) = m 1 Σ + 12 Σ (x − m 2 ) ⎪ 22 2 ⎨ (2) T ⎪ Σ 1|2 (x ) Σ = 11 Σ − 12 Σ − 1 Σ 2 22 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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