Applied Statistics Using SPSS, STATISTICA, MATLAB and R

A.8.2 Moments
A.8 Multivariate Distributions 425
The moments of multivariate random variables are a generalisation of the previous
definition for single variables. In particular, for bivariate distributions, we have the
central moments:
k
j
m = Ε[
( X − µ ) ( Y − µ ) ] . A. 44
kj
x
The following central moments are worth noting:
y
m20 = 2 σ X : variance of X; m02 = 2 σ Y : variance of Y;
m = Ε XY − µ µ .
m11 ≡ σXY= σYX: covariance of X and Y, with 11 [ ] X Y
For multivariate d-dimensional distributions we have a symmetric positive
definite covariance matrix:
⎡ 2
σ
⎤
1 σ 12 K σ 1d
⎢
2 ⎥
= ⎢σ
21 σ 2 K σ 2d
Σ ⎥ . A. 45
⎢ K K K K ⎥
⎢
⎥
2
⎢⎣
σ d1
σ d 2 K σ d ⎥⎦
The correlation coefficient, which is a measure of linear association between X
and Y, is defined by the relation:
σ XY
ρ ≡ ρ XY = . A. 46
σ . σ
X
Y
Properties of the correlation coefficient:
i. –1 ≤ ρ ≤ 1;
ii. ρ XY = ρYX
;
iii. ρ = ±1 iff ( Y − µ Y ) / σ Y = ± ( X − µ X ) / σ X ;
iv. ρ aX + b,cY + d = ρ XY , ac > 0; ρ aX + b,cY + d = −ρ
XY , ac < 0.
If m11 = 0, the random variables are said to be uncorrelated. Since
Ε [ XY ] = Ε[
X ] Ε[
Y ] if the variables are independent, then they are also
uncorrelated. The converse statement is not generally true. However, it is true in
the case of normal distributions, where uncorrelated variables are also independent.
The definitions of covariance and correlation coefficient have a straightforward
generalisation for the d-variate case.
A.8.3 Conditional Densities and Independence
Assume that the bivariate random vector [X, Y] has a density function f(x, y). Then,
the conditional distribution of X given Y is defined, whenever f(y) ≠ 0, as: