Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 125

So, X e Y independent !

EXY ðÞ ¼ EX ðÞ EY ðÞ;

Cov X ð ; Y Þ ¼ 0 ;

q XY ¼ 0

and

VX ð þ Y Þ ¼ VX ðÞþ VY ðÞ:

A.6 Conditional Distributions

A more complete information on the dependence between random variables is
given by the conditional expectations.
For any p and any random variables X and Y de ∈

R for which p[X=x] > 0, the expected value of Y with respect to p conditional on
the event {s

ned on (S, ʛ ) and any x

S | X(s) = x}, that means with respect to the probability p(|[X = x])
can be computed. It is denoted E(Y|X = x).
If X is a discrete random variable, a function Y |X with domain S and codomain
R that for each s with p[X = X(s)] > 0 assigns Y |X (s) = E(Y|X = X(s)) is a random
variable and its expected value E(EY |X ) equals the expected value EY of Y.
More generally, for any A

∈

, the expected value of the restriction of Y |X to A
coincides with the expected value of the restriction of Y to A, that means,
EY j X 1 A

∈ ʛ

¼

EY1 A ð Þ:

Or extending yet a little more, for any random variable g(X, Y)
EgX

:

ð ð ; Y Þ Þ ¼ EgX ; EY j X

nition of conditional expectation.
For any X and Y and any Z that is constant in any set of

This property is employed to extend the de

where X is constant and

Λ

satis es E(g(X, Z)) = E(g(X, Y)), Z is a conditional expectation of Y given X.

This can be extended to conditioning on a vector of random variables. It is
enough to replace in the above formulation the random variable X by a vector X =
(X 1 , … ,X n ) of random variables X 1 , … ,X n .

Replacing Y by 1 [Y ∈ A] , for each event A, this de nition of conditional

expectation can be used to de ne a distribution of Y conditional on X = x, denoted

by p Y (|X = x) such that a conditional expectation of Y given X can be obtained
computing, for each x, the expectation of Y with respect to this distribution of Y
conditional on X = x.
In the discrete case, the distribution of Y conditional on X = x is given by
p Y A ½ j X ¼ x ¼ p ð X ¼ x ; Y 2 A Þ= pX ð ¼ x Þ:

For continuous random vectors, the conditional density of Y given X is, for each
real x, the density of the conditional distribution of Y given X = x, that is the real