{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# STATSLIDE7 - MULTIVARIATE DISTRIBUTIONS Each element a...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 x i . The expected value E ( x ) of the vector x = [ x 1 , x 2 , . . . , x n ] 0 is the vector containing the expected values of the elements: E ( x ) = [ E ( x 1 ) , E ( x 2 ) , . . . , E ( x n )] 0 = [ μ 1 , μ 2 , . . . , μ n ] 0 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

STATSLIDE7 - MULTIVARIATE DISTRIBUTIONS Each element a...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online