MVN_density_cf

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 μ and covariance matrix Λ given by μ = μ x μ y and Λ = σ 2 x ρσ x σ y ρσ x σ x σ 2 y . Note that ρ = corr( X, Y ) in this notation. The characteristic function of X is ϕ X ( t ) = exp iμ x t 1 + iμ y t 2- 1 2 ( σ 2 x t 2 1 + 2 ρσ x σ y t 1 t 2 + σ 2 y t 2 2 ) which written in matrix notation is ϕ X ( t ) = exp i t μ- 1 2 t Λ t . The density function of X is f X ( x, y ) = 1 2 πσ x σ y p 1- ρ 2 exp- 1 2(1- ρ 2 ) x- μ x σ x 2- 2 ρ ( x- μ x )( y- μ y ) σ x σ y + y- μ y σ y 2 ! which written in matrix notation is f X ( x, y ) = 1 2 π 1 √ detΛ exp- 1 2 ( x- μ ) Λ- 1 ( x- μ ) . Of course, there are some noticeable similarities between these two functions. In particular, if μ = (0 , 0) , then ϕ X ( t ) = exp- 1 2 ( σ 2 x t 2 1 + 2 ρσ x σ y...
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This note was uploaded on 03/26/2012 for the course STAT 351 taught by Professor Michaelkozdron during the Fall '08 term at University of Regina.

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

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