Lecture7 - MULTIVARIATE DISTRIBUTIONS An n-dimensional...

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MULTIVARIATE DISTRIBUTIONS An n -dimensional random vector x =[ x 1 ,x 2 ,...,x n ] 0 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 2 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 0 1 0 2 ] 0 , where x 0 1 x 1 2 m ] 0 and x 0 2 x m +1 m +2 n ] 0 . (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 Z x 2 f ( x 1 2 ) dx 2 = Z x n ··· Z x n +1 f ( x 1 2 m m +1 , 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 2 ) f ( x 2 ) .
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Lecture7 - MULTIVARIATE DISTRIBUTIONS An n-dimensional...

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