Engineering Analysis/Expectation and Entropy
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.
A moment is a value that contains some information about the random variable. The n-moment of a random variable is defined as:
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.
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.
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
the mean and variance of a random variable can be related by:
This is an important function, and we will use it later.
the entropy of a random variable is defined as: