Engineering Analysis/Expectation and Entropy

Expectation

The expectation operator of a random variable is defined as:

$E[x]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x{f}_X(x)dx$

This operator is very useful, and we can use it to derive the moments of the random variable.

Moments

A moment is a value that contains some information about the random variable. The n-moment of a random variable is defined as:

$E[x^n]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^n{f}_X(x)dx$

Mean

The mean value, or the "average value" of a random variable is defined as the first moment of the random variable:

$E[x]={\mu }_X=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x{f}_X(x)dx$

We will use the Greek letter μ to denote the mean of a random variable.

Central Moments

A central moment is similar to a moment, but it is also dependent on the mean of the random variable:

$E[(x-{\mu }_X{)}^n]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(x-{\mu }_X{)}^n{f}_X(x)dx$

The first central moment is always zero.

Variance

The variance of a random variable is defined as the second central moment:

$E[(x-{\mu }_X{)}^2]={\sigma }^2$

The square-root of the variance, σ, is known as the standard-deviation of the random variable

Mean and Variance

the mean and variance of a random variable can be related by:

${\sigma }^2={\mu }^2+E[x^2]$

This is an important function, and we will use it later.

Entropy

the entropy of a random variable $X$ is defined as:

$H[X]=E\left[\frac{1}{p(X)}\right]$