STATSLIDE7 - MULTIVARIATE DISTRIBUTIONS Each element a...

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MULTIVARIATE DISTRIBUTIONS Each element a random vector x = [ x 1 , x 2 , . . . , x n ] 0 describes an aspect of a statistical outcome. We write x ∈ R n to signify that x is a point in a real n -space. A function f ( x ) = f ( x 1 , x 2 , . . . , x n ) assigning a probability measure to every point in R n is called a multivariate p.d.f. Consider the partitioned vector x = x 1 x 2 = [ x 0 1 , x 0 2 ] 0 , wherein x 0 1 = [ x 1 , x 2 , . . . , x m ] 0 and x 0 2 = [ x m +1 , x m +2 , . . . , x n ] 0 . The marginal p.d.f, of x 1 is f ( x 1 ) = Z x 2 f ( x 1 , x 2 ) dx 2 = Z x n · · · Z x m +1 f ( x 1 , . . . , x m , x m +1 , · · · , x n ) dx m +1 , . . . , d n , whereas the conditional p.d.f of x 1 given x 2 is f ( x 1 | x 2 ) = f ( x ) f ( x 2 ) = f ( x 1 , x 2 ) f ( x 2 ) . 1
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The expected value of the i th element of x is E ( x i ) = Z x x i f ( x ) dx = Z x n · · · Z x 1 x i f ( x 1 , . . . , x n ) dx 1 , . . . , dx n = Z x i x i f ( x i ) dx i , where f ( x i ) is the marginal distribution of
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This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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STATSLIDE7 - MULTIVARIATE DISTRIBUTIONS Each element a...

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