Engineering Analysis/Expectation and Entropy

From Wikibooks, open books for an open world
Jump to: navigation, search

Expectation[edit]

The expectation operator of a random variable is defined as:

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

Moments[edit]

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

Mean[edit]

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

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

Central Moments[edit]

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

The first central moment is always zero.

Variance[edit]

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

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

Mean and Variance[edit]

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

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

Entropy[edit]

the entropy of a random variable is defined as: