moment

Moments

Given a random variable $X$, the $k$th moment of $X$ is the value $E[X^k]$, if the expectation exists.

Note that the expected value is the first moment of a random variable, and the variance is the second moment minus the first moment squared.

The $k$th moment of $X$ is usually obtained by using the moment generating function.

Central moments

Given a random variable $X$, the $k$th central moment of $X$ is the value $E\left[{(X-E[X])}^k\right]$, if the expectation exists. It is denoted by ${\mu }_k$.

Note that the ${\mu }_1=0$ and ${\mu }_2=Var[X]={\sigma }^2$. The third central moment divided by the standard deviation cubed is called the skewness $\tau$:

$$\tau =\frac{{\mu }_3}{{\sigma }^3}$$

The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution is with respect to its mode. A non-zero $\tau$ means there is some degree of skewness in the distribution. For example, $\tau >0$ means that the distribution has a longer positive tail.

The fourth central moment divided by the fourth power of the standard deviation is called the kurtosis $\kappa$:

$$\kappa =\frac{{\mu }_4}{{\sigma }^4}$$

The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution. The standard normal distribution has $\kappa =3$. means that the distribution is “flatter” than then standard normal distribution, or platykurtic. On the other hand, a distribution with $\kappa >3$ can be characterized as being more “peaked” than $N(0,1)$, or leptokurtic.

Title	moment
Canonical name	Moment
Date of creation	2013-03-22 11:53:54
Last modified on	2013-03-22 11:53:54
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60-00
Classification	msc 62-00
Classification	msc 81-00
Defines	central moment
Defines	skewness
Defines	kurtosis
Defines	platykurtic
Defines	leptokurtic