Probability theory MOC

Moment

Let be a real random variable with mean and variance . For any , we define #m/def/prob

• the th moment of as
• the th central moment of as
• the th standardized moment¹ of as

if said quantities exist. Additionally, for a -valued discrete random variable

• the th factorial moment of is

where is the Probability generating function.

Examples

• The first central moment is the mean
• The second central moment is the variance
• The third standardized moment is the Skewness
• The fourth standardized moment is a shifted version of the Excess kurtosis

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