Engineering Analysis/Expectation and Entropy

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[edit] 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.

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

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

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.

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.

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

E [(x - μ X) 2] = σ 2

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

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

σ 2 = μ 2 + E [x 2]

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

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

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