Expectation of a Random Variable
Overview
The expectation of a random variable is simply defined to be the
integral
of that variable.
Definitions
The expected value of a random variable {% X %} is defined to be
{% \mathbb{E}(X) = \int x f(x) dx %}
where {% f(x) %} is the probability density function.
Restated using
measure theory
notation
{% \mathbb{E}(X) = \int X(\omega) d \mathbb{P}(\omega) %}
for a discrete variable, it is defined as
{% \mathbb{E}(X) = \sum x \times prob(x) %}
Properties
If {% X %} is non random, then
{% \mathbb{E}(X) = X %}
Also, the expectation is linear
{% \mathbb{E}(aX + bY) = a\mathbb{E}(X) + b\mathbb{E}(Y) %}
If {% X %} and {% Y %} are
independent,
then
{% \mathbb{E}(XY) = \mathbb{E}(X) \times \mathbb{E}(Y) %}
