# Engineering Analysis/Expectation and Entropy

## Expectation

The expectation operator of a random variable is defined as:

$E[x] = \int_{-\infty}^\infty x f_X(x)dx$

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:

$E[x^n] = \int_{-\infty}^\infty x^n f_X(x)dx$

### Mean

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

$E[x] = \mu_X = \int_{-\infty}^\infty x f_X(x)dx$

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 dependant on the mean of the random variable:

$E[(x - \mu_X)^n] = \int_{-\infty}^\infty (x - \mu_X)^n f_X(x)dx$

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:

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