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Random Variable Measure Positive Valency Distribution Expected Value Variance Normal Random Variable Mode Moment Generating Function Standard Deviation Fourth Power

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Moment

Moments

Given a random variable , the th moment of is the value , 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 th moment of is usually obtained by using the moment generating function.

Central moments

Given a random variable , the th central moment of 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 skewness $\tau$ :

$\displaystyle \tau=\frac{\mu_3}{\sigma^3}$

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 $\kappa$ :

$\displaystyle \kappa=\frac{\mu_4}{\sigma^4}$

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 platykurtic. On the other hand, a distribution with $\kappa>3$ can be characterized as being more “peaked” than

, or leptokurtic.

1	Random Variable
1	Measure
1	Positive
2	Valency
3	Variance
3	Expected Value
3	Distribution
4	Mode
4	Normal Random Variable
7	Standard Deviation
7	Moment Generating Function
9	Fourth Power