This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Statistics 351 (Fall 2007) The Density and Characteristic Function Definitions of Multivariate Normality Suppose that the random vector X = ( X, Y ) has a multivariate normal distribution with mean vector and covariance matrix given by = x y and = 2 x x y x x 2 y . Note that = corr( X, Y ) in this notation. The characteristic function of X is X ( t ) = exp i x t 1 + i y t 2 1 2 ( 2 x t 2 1 + 2 x y t 1 t 2 + 2 y t 2 2 ) which written in matrix notation is X ( t ) = exp i t  1 2 t t . The density function of X is f X ( x, y ) = 1 2 x y p 1 2 exp 1 2(1 2 ) x x x 2 2 ( x x )( y y ) x y + y y y 2 ! which written in matrix notation is f X ( x, y ) = 1 2 1 det exp 1 2 ( x )  1 ( x ) . Of course, there are some noticeable similarities between these two functions. In particular, if = (0 , 0) , then X ( t ) = exp 1 2 ( 2 x t 2 1 + 2 x y...
View Full
Document
 Fall '08
 MichaelKozdron
 Statistics, Covariance, Normal Distribution, Probability, Variance

Click to edit the document details