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Review2_Study_Guide

Review2_Study_Guide - ORIE 3500/5500 Fall Term 2008 Review...

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ORIE 3500/5500 Fall Term 2008 Review 2 Multivariate Normal A vector ( X 1 , . . . , X k ) is said to follow a multivariate normal distribution if any linear combination of the components, e.g. Y = a 1 X 1 + · · · + a k X k for some constants a 1 , . . . , a k , has normal distribution. The parameters of the multivariate normal distribution are (a) the mean vector μ and (b) covariance matrix Σ μ = μ 1 μ 2 . . . μ k , Σ = σ 11 σ 12 · · · σ 1 k σ 21 σ 22 · · · σ 2 k . . . . . . . . . σ k 1 σ k 2 · · · σ kk , and the notation is X = X 1 X 2 . . . X k N k [ μ, Σ ] . The interpretation of the parameters are μ i = E ( X i ) , σ ij = cpv ( X i , X j ) , 1 i, j k. Three important things to remember: 1. Suppose Y = ( Y 1 , . . . , Y r ) T be such that each com ponent of Y is some linear cpmbination of the components of X , e.g. Y = Y 1 Y 2 . . . Y r = a 11 X 1 + · · · + a 1 k X k a 21 X 1 + · · · + a 2 k X k .

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