Engineering Analysis/Expectation and Entropy

Expectation

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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

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

 

Mean

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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

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

 

The first central moment is always zero.

Variance

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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

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

 

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

Entropy

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the entropy of a random variable   is defined as: