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Computer Science
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CSCC37H3
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Lecture

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Computer Science

CSCC37H3

Malcolm Mac Kinnon

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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