Moments

Probability Moments

Moments are a set of statistical parameters used to measure the characteristics of a distribution’s shape. For a random variable $X$, the $n$-th moment is the expected value of the $n$-th power of $X$.

Raw Moments (About the Origin)

Raw moments are calculated relative to zero. The $n$-th raw moment is denoted as ${\mu }_n^{\prime }$:

$${\mu }_n^{\prime }=E[X^n]$$

The mean ($\mu$), representing the “center of mass” of the distribution.

Central Moments (About the Mean)

Central moments describe the shape of the distribution independent of its location. The $n$-th central moment is denoted as ${\mu }_n$:

$${\mu }_n=E[(X-\mu {)}^n]$$

• 1st Central Moment: Always equals $0$.
• 2nd Central Moment: The Variance (${\sigma }^2$), measuring the spread or scale.
• 3rd Central Moment: Used to calculate Skewness (asymmetry).
• 4th Central Moment: Used to calculate Kurtosis (thickness of tails).

Standardized Moments

To compare different distributions regardless of scale, moments are normalized by the standard deviation $\sigma$:

${\tilde{\mu }}_n=\frac{{\mu }_n}{{\sigma }^n}$

Moment	Name	Interpretation
${\mu }_1^{\prime }$	Mean	Central location.
${\mu }_2$	Variance	Width/Dispersion.
${\tilde{\mu }}_3$	Skewness	Direction and strength of the tail.
${\tilde{\mu }}_4$	Kurtosis	Frequency of outliers (“Peakedness”).

Moment Generating Function (MGF)

The MGF is a powerful tool that “encapsulates” all moments into a single function:

$$M_X(t)=E[e^{tX}]$$

To extract the $n$-th raw moment, differentiate the MGF $n$ times and evaluate at $t=0$:

$${\mu }_n^{\prime }={\frac{d^n}{d{t}^n}{M}_X(t)|}_{t=0}$$