MVN_density_cf - Statistics 351 (Fall 2007) The Density and...

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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...
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MVN_density_cf - Statistics 351 (Fall 2007) The Density and...

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