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.
Given a random variable , the th central moment of is the value , if the expectation exists. It is denoted by .
Note that the and . The third central moment divided by the standard deviation cubed is called the skewness :
The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution is with respect to its mode. A non-zero means there is some degree of skewness in the distribution. For example, 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 :
The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution. The standard normal distribution has . means that the distribution is “flatter” than then standard normal distribution, or platykurtic. On the other hand, a distribution with can be characterized as being more “peaked” than , or leptokurtic.
|Date of creation||2013-03-22 11:53:54|
|Last modified on||2013-03-22 11:53:54|
|Last modified by||CWoo (3771)|