Moment 2001-10-26 03:04:14 2006-09-23 12:15:50 Definition central moment skewness kurtosis platykurtic leptokurtic \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \textit{Moments}\\ \par Given a random variable $X$, the \textbf{$k$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.\\ \par The $k$th moment of $X$ is usually obtained by using the moment generating function. \par \par \textit{Central moments}\\ \par Given a random variable $X$, the \textbf{$k$th central moment} of $X$ is the value $E\big[(X-E[X])^k\big]$, 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 \emph{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. \par The fourth central moment divided by the fourth power of the standard deviation is called the \emph{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 $\kappa=3$. $\kappa<3$ means that the distribution is ``flatter'' than then standard normal distribution, or \emph{platykurtic}. On the other hand, a distribution with $\kappa>3$ can be characterized as being more ``peaked'' than $N(0,1)$, or \emph{leptokurtic}.