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Multinormal

# Multinormal - Page 1 Chapter 12 Multivariate normal...

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Page 1 Chapter 12 Multivariate normal distributions The multivariate normal is the most useful, and most studied, of the standard joint dis- tributions in probability. A huge body of statistical theory depends on the properties of fam- ilies of random variables whose joint distribution is at least approximately multivariate nor- mal. The bivariate case (two variables) is the easiest to understand, because it requires a minimum of notation; vector notation and matrix algebra becomes necessities when many random variables are involved. The general bivariate normal is often used to model pairs of dependent random vari- ables, such as : the height and weight of an individual; or (as an approximation) the score a student gets on a final exam and the total score she gets on the problem sets; or the heights of father and son; and so on. Many fancy statistical procedures implicitly require bivariate (or multivariate, for more than two random variables) normality. Bivariate normal The most general bivariate normal can be built from a pair of independent random vari- ables, X and Y , each distributed N ( 0 , 1 ) . For a constant ρ with - 1 < ρ < 1, define random variables U = X and V = ρ X + p 1 - ρ 2 Y That is, ( U , V ) = ( X , Y ) A where A = 1 ρ 0 p 1 - ρ 2 Notice that E U = E V = 0, var ( V ) = ρ 2 var ( X ) + ( 1 - ρ 2 ) var ( Y ) = 1 = var ( U ), and cov ( U , V ) = ρ cov ( X , X ) + p 1 - ρ 2 cov ( X , Y ) = ρ. Consequently, correlation ( U , V ) = cov ( U , V )/ p var ( U ) var ( V ) = ρ From Chapter 10, the joint density for ( U , V ) is 1 | det A | f ( ( u , v) A - 1 ) , where f ( x , y ) = 1 2 π exp - x 2 + y 2 2 all x , y The matrix A has determinant p 1 - ρ 2 and inverse A - 1 = p 1 - ρ 2 - ρ 0 1 / p 1 - ρ 2 Statistics 241: 16 November 1997 c David Pollard

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Chapter 12 Multivariate normal distributions Page 2 If ( x , y ) = ( u , v) A - 1 then x 2 + y 2 = ( u , v) A - 1 ( A - 1 ) 0 ( u , v) 0 = ( u , v) 1 - ρ - ρ 0 ( u , v) 0 /( 1 - ρ 2 ) = u 2 - 2 ρ u v + v 2 1 - ρ 2 Thus U and V have joint density 1 2 π p 1 - ρ 2 exp - u 2 - 2 ρ u v + v 2 2 ( 1 - ρ 2 ) for all u , v. The joint distribution is sometimes called the standard bivariate normal distribution standard bivariate normal with correlation ρ . The symmetry of ψ in u and v implies that V has the same marginal distribution as U , that is, V is also N ( 0 , 1 ) distributed. The calculation of the marginals densities involves the same integration for both variables.
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Multinormal - Page 1 Chapter 12 Multivariate normal...

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