Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 126

function f Y|X=x such that, for F Y|X=x denoting the cumulative distribution function
of the conditional distribution of Y given the event [X = x],

Z

y
1

F Y j X¼x y ðÞ ¼ f Y j X¼x ð u Þ du :

A better understanding of the information given by the conditional expectation on
the correlation between different variables is obtained by computing the variance of
the conditional expectation as the variance of a random variable whose variation is
limited by eliminating all dispersion of Y within any set determined by a

xed
value of X. The variance of Y can be decomposed into the sum of this variance of
the conditional expectation and the expectation of another variable, the variance of
the distribution of Y conditional on X.

VY ¼ V ð EY ð j X Þþ E ð VY ð j X Þ;

for V(Y|X) denoting the function that associates to each x the variance of the
distribution with cdf F Y|X=x .

A.7 Basic Distributions

This section brings examples of probabilities of each kind that will be useful in the
development of the probabilistic determination of preferences.

A.7.1 Examples of Discrete Probability Distributions

A.7.1.1 The Bernoulli Distribution

The simplest sample space is that of the occurrence or not of a well speci ed fact, like

success in an experiment or acceptance of a proposal. The space of events has then
only 4 elements: {{yes}, {no},

, S = {yes, no}}. A random variable has a Bernoulli
distribution when its range has only two values 1 and 0, 1 associated to the occurrence
of a given event of probability q and 0 associated to its complement. Thus

Φ

p X x ðÞ ¼ q for x ¼ 1 ; p X x ðÞ ¼ 1 q for x ¼ 0 and p X x ðÞ ¼ 0 for any other real x :

This implies that the cdf of X assumes the value 0 in R − ,1 − q in the interval [0,1)

and 1 otherwise.
The expected value of a random variable with the Bernoulli distribution 1 A is p(A).
Since 1 2 =1 A , the variance is
V1 ðÞ ¼ pA ðÞ ð 1 pA ðÞ Þ:

Thus, for small p(A), V(1 A ) is slightly smaller than E(1 A ).