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Lecture7 - MULTIVARIATE DISTRIBUTIONS An n-dimensional...

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MULTIVARIATE DISTRIBUTIONS An n -dimensional random vector x = [ x 1 , x 2 , . . . , x n ] is an ordered set of n random variables, each of which describes some aspect of a statistical outcome. We write x ∈ R n to signify that x is a point in a real n -dimensional space. A function f ( x ) = f ( x 1 , x 2 , . . . , x n ) that assigns 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 . To conserve space, we prefer to write this as a transposed row vector x = [ x 1 , x 2 ] , where x 1 = [ x 1 , x 2 , . . . , x m ] and x 2 = [ x m +1 , x m +2 , . . . , x n ] . (Notice that, in this notation, the subvector x 1 and the leading scalar element x 1 of the vector x are represented by the same symbols. However, these entities can be distinguished by their context.) The marginal p.d.f, of x 1 is f ( x 1 ) = x 2 f ( x 1 , x 2 ) dx 2 = x n · · · x n +1 f ( x 1 , x 2 , . . . , 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 ) .
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