Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 124

To make possible comparison by controlling for the effect of scales, instead of
covariance is employed the correlation coef cient.

The correlation coef cient between the random variables X and Y is

ð X ; Y Þ ¼ Cov X ð ; Y Þ= r ð ðÞr X ðÞ Y Þ

The following are the properties that make the coef cient of correlation

advantageous relatively to the covariance:

0 q j ð X ; Y Þ j 1

Y ¼ cX ! q j ð X ; Y Þ j ¼ 1 for c ¼ 0 ;

q ð X ; Y Þ ¼ þ 1ifc 0

[

and

q ð X ; Y Þ ¼ 1ifc \ 0 :

$

The random variables X and Y are independent

p ½ a X b ; c Y d ¼ p ½ a X bp ½ c Y d

for every real numbers a, b, c and d.
For X and Y discrete random variables, this de nition may be put in simpler

terms:

X e Y are independent $ pX ½ ¼ a ; Y ¼ b

¼ pX ½ ¼ a pY ½ ¼ b for every real a and b :

So the random variables 1 A e1 B are independent $ the events A and B are

independent.
For continuous variables, X and Y are independent if and only if

f XY ¼ f X f Y :

X and Y are independent if and only if

EgX ½ ðÞ hY ðÞ ¼ EgX ½ ðÞ EhY ½ ðÞ for all real functions g and h ;

i.e., the expectation of the product of a function of X by a function of Y with respect
to the joint distribution of X and Y is equal to the product of the expectations of the
two random variables computed separately.
For continuous random variables X and Y with joint density f XY and marginal
densities f X and f y , this means:
ZZ gx Z gx

dx Z hx

ðÞ hy ðÞ f XY x ðÞ ; y dxdy ¼ ðÞ f X x ðÞ ðÞ f Y y ðÞ dy :