Signals and Systems/Probability Operations

Expected Value Operator

The Expected value operator is a linear operator that provides a mathematical way to determine a number of different parameters of a random distribution. The downside of course is that the expected value operator is in the form of an integral, which can be difficult to calculate.

The expected value operator will be denoted by the symbol:

$\mathbb {E} [\cdot ]$

For a random variable X with probability density f_x, the expected value is defined as:

$\mathbb {E} [X]=\int _{-\infty }^{\infty }xf_{X}(x)dx$.

provided the integral exists.

The Expectation of a signal is the result of applying the expected value operator to that signal. The expectation is another word for the mean of a signal:

$\mu _{x}=\mathbb {E} [X]$

Moments

The expected value of the N-th power of X is called the N-th moment of X or of its distribution:

$\mathbb {E} [X^{N}]=\int _{-\infty }^{\infty }x^{N}f_{X}(x)dx$.

Some moments have special names, and each one describes a certain aspect of the distribution.

Central Moments

Once we know the expected value of a distribution, we know its location. We may consider all other moments relative to this location and calculate the N^th moment of the random variable X - E[X]; the result is called the N^th central moment of the distribution. Each central moment has a different meaning, and describes a different facet of the random distribution. The N-th central moment of X is:

$\mathbb {E} [(X-\mathbb {E} [X])^{N}]=\int _{-\infty }^{\infty }(x-\mathbb {E} [X])^{N}f_{X}(x)dx$.

For sake of simplicity in the notation, the first moment, the expected value is named:

$\mathbb {E} [X]=\mu _{X}$,

The formula for the N-th central moment of X becomes then:

$\mathbb {E} [(X-\mu _{X})^{N}]=\int _{-\infty }^{\infty }(x-\mu _{X})^{N}f_{X}(x)dx$.

It is obvious that the first central moment is zero:

$\mathbb {E} [(X-\mu _{X})]=0$

The second central moment is the variance,

$\mathbb {E} [(X-\mu _{X})^{2}]=\sigma ^{2}$

Variance

The variance, the second central moment, is denoted using the symbol σ_x², and is defined as:

$\sigma _{x}^{2}=\mathrm {var} (X)=\mathbb {E} [X-\mathbb {E} [X]]^{2}=\mathbb {E} [X^{2}]-\mathbb {E} [X]^{2}$

Standard Deviation

The standard deviation of a random distribution is the square root of the variance, and is given as such:

$\sigma _{x}={\sqrt {\sigma _{x}^{2}}}$

Time-Average Operator

The time-average operator provides a mathematical way to determine the average value of a function over a given time range. The time average operator can provide the mean value of a given signal, but most importantly it can be used to find the average value of a small sample of a given signal. The operator also allows us a useful shorthand way for taking the average, which is used in many equations.

The time average operator is denoted by angle brackets (< and >) and is defined as such:

$\langle f(t)\rangle ={\frac {1}{T}}\int _{0}^{T}f(t)dt$