notes7 - Stat 430/Math468 Notes#7 The Multivariate Normal...

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Stat 430/Math468 – Notes #7 Chapter 4 The Multivariate Normal Distribution (Continued) If a random vector has multivariate normal distribution, then it has the following properties: X 1. Linear combinations of the components of are normally distributed. X 2. All subsets of the components of have (multivariate) normal distribution. X 3. Uncorrelated components are independent. 4. The conditional distributions of the components are (multivariate) normal. Formally, we have the following results: Result: 1. If and , then . ~( , p N X μΣ ) 1 ( ,. .., )' p aa = a '~( ' , ' ) N aX a μ a Σ a 2. If for every ' , N a μ a Σ a 1 ( ,. .., p = a , then . , p N X ) p The proof needs multivariate moment generating function (skipped). The two results together can be considered as a characteristic (definition) of multivariate normal distribution. Example: Let , then what is and what is its distribution? The i-th element is 1 others are 0's ( 0,0,. ..0,1,0. ..,0 )', 1,2,. .., i i == a 1442 4 43 ' i 1
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Result: Suppose . ~( , p N X μΣ ) ) ) 1. If , then .
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notes7 - Stat 430/Math468 Notes#7 The Multivariate Normal...

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