Probability theory MOC

Moment

Let 𝑋 :𝜉 →ℝ be a real random variable with mean 𝜇 and variance 𝜎2. For any 𝑛 ∈ℕ, we define #m/def/prob

• the 𝑛th moment of 𝑋 as 𝔼⁡[𝑋𝑛]
• the 𝑛th central moment of 𝑋 as 𝔼⁡[(𝑥 −𝜇)𝑛]
• the 𝑛th standardized moment¹ of 𝑋 as 𝔼⁡[(𝑋−𝜇𝜎)𝑛]

if said quantities exist. Additionally, for a ℕ0-valued discrete random variable

• the 𝑛th factorial moment of 𝑋 is 𝔼⁡[∏𝑛−1𝑗=0(𝑋−𝑗)] =𝑔(𝑘)𝑋(1)

where 𝑔𝑋 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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