Real-Valued Random Variable Variance
Definition
The variance of a square-integrable real-valued random variable is the expectation of its square less its expectation squared.
Notation
Suppose $f: X \to \R $ is a random variable
on a probability space $(X, \mathcal{A} , P)$.
We denote the variance of $f$ by $\var f$, so
that
\[
\var f L= \E (f^2) - (\E (f))^2.
\]
Results
If a random variable on a probability space is
square integrable then it is integrable.
The $L^p$ spaces are nested
for finite measures.
The variance of a square-integrable real-valued
random variable is the expectation of the square
of the difference between the random variable
and its expectation.