mvnormal.pdf

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Department
Computer Science
Course
CSCC37H3
Professor
Malcolm Mac Kinnon
Semester
Summer

Description
3.5 Multivariate Normal Distribution • Defintion: A random vector (X ,X ,...,X ) is said to have the multivariate normal distribution with 1 2 n parameters ▯ ▯ ▯1 ▯12 ··· ▯1n ▯ 2 ▯ ▯▯ T ▯ ▯ 21 ▯2 ··· ▯2n ▯ µ =( µ 1...,µn) and ▯ = ▯ . ▯ ▯ . ▯ ▯ ▯ ··· ▯2 n1 n2 n if it has joint probability density function given by 1 1 ▯▯ ▯▯ T ▯1 ▯▯ ▯▯ n fX1,...nXx1,...,n )= n 1/2exp{▯ ( x ▯ µ ) ▯ ( x ▯ µ )} for (1 ,...,n ) ▯ R ((2▯) det▯) 2 and we write (X ,X ,...,X ) ▯ N ( µ, ▯) 1 2 n n • Again µ is a location parameter and ▯ is a spread parameter this time in R . • ▯ is symmetric. For exampleij= ▯ji • We’ll assume throughout that ▯ is positive definite. For example a ▯ a> 0 for all vectors a ▯= 0. • Note that ( x ▯ µ ) ▯1( x ▯ µ ) is the squared distance of x to µ in ▯ units. • Notation: When n = 2 and for the random vector ▯▯ ▯ ▯ 2 ▯▯ (X,Y ) ▯ N µX , ▯X ▯XY 2 µY ▯XY ▯Y we will often write ▯ =Y . ▯X ▯Y For Example: T Denote by (X,Y ) the random vector where X is the weight of a B52 student in pounds and Y equals the height of the same B52 student in feet. ▯▯ ▯ ▯ ▯▯ µX ▯ 2 ▯XY Suppose that (X,Y ) ▯ N2 , X 2 and letting ▯ =XY . µY ▯XY ▯Y ▯X▯Y A well known result in linear algebra implies ▯ 2 ▯ ▯ 2 ▯ ▯1 1 ▯Y ▯▯ XY 1 ▯Y ▯▯▯X▯ Y ▯ = ▯ ▯ ▯ ▯ 2 ▯▯XY ▯2 = ▯ ▯ (1 ▯ ▯ ) ▯▯▯ X Y ▯2 . X Y XY X X Y X By Substitution and some algebra it follows 49 ▯ ▯▯ ▯ ▯▯ ▯▯ T ▯1 ▯▯ ▯▯ 1 ▯Y ▯▯▯X▯ Y x ▯ X ( x ▯ µ ) ▯ ( x ▯ µ )=( x ▯Xµ ,y ▯Yµ )2 2 2 ▯▯▯ ▯ ▯2 y ▯ µ ▯ X Y1 ▯ ▯ ) X Y X Y ▯ 2(x ▯ µ ) ▯ 2▯▯ ▯ (x ▯ µ )(y ▯ µ )+ ▯ (y ▯ µ ) = Y X X Y X Y X Y ▯X▯ Y1 ▯ ▯ ) ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ 1 x ▯ X 2 x ▯ µX y ▯ Y y ▯ µY 2 = ▯ 2▯ + . 1 ▯ ▯2 ▯ X ▯X ▯Y ▯Y Putting all of this together it follows that ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ 1 1 x ▯ X 2 x ▯ µX y ▯ Y y ▯ µY 2 fXY(x,y)= ▯ exp ▯ ▯ 2▯ + . 2▯ ▯X▯Y(1 ▯ ▯ ) 2(1 ▯ ▯ ) ▯X ▯X ▯Y ▯Y • Visually: • We can investigate those ordered pairs in the support that have equal density in probability. • For example and for fixed C> 0 ▯ ▯▯ ▯2 ▯ ▯▯ ▯ ▯ ▯2▯▯ 1 1 x ▯ µX x ▯ µX y ▯ Y y ▯ Y C = ▯ 2 2 exp ▯ 2 ▯ 2▯ + 2▯ ▯X▯ Y1 ▯ ▯ ) 2(1 ▯ ▯ ) ▯X ▯X ▯Y ▯Y if and only if ▯ ▯ 2 ▯ ▯▯ ▯ ▯ ▯2 ▯ ▯ ▯ x ▯ µX ▯ 2▯ x ▯ X y ▯ µY + y ▯ Y = ▯2(1 ▯ ▯ )log C · 2▯▯ ▯ (1 ▯ ▯ ) ▯X ▯X ▯Y ▯Y X Y which is the equation of an ellipse. ▯XY • In the situation where X Y = 0, the axes of each ellipse of constant density are in the same direction as the co-ordinate axes. • In the situation where ▯ = ▯= 0, the axes of each ellipse of constant density are in the direction of the ▯X▯Y eigenvectors of . 50 • In the situation where ▯ =XY ▯= 0 we conclude thatXY is the parameter that determines the orientation ▯X▯Y of the ellipse. • To describe the orientation of the ellipse usXYg we note that probability density is concentrated along the line ▯ y ▯ Y = XY (x ▯ µX) ▯X • There’s a sometimes preferrable definition of the multivariate normal distribution. T • Definition: A random vector (X1,X 2...,X n is said to have the multivariate normal distribution with parameters ▯ 2 ▯ ▯1 ▯12 ··· ▯1n ▯ ▯ 21 ▯2 ··· ▯2n ▯
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