STATISTICS Course

Moments

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Moments are the expected values, the variance, Skewness, and Kurtosis. As you should have seen in the probability course,

  • $\mathbb{E}(X) = \mathbb{E}[X^1]$ is the first moment
  • $V(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2$ is the second centered moment (centered because it's the second moment minus the expected value)

3rd and 4th moments

And we may also use now, the 3rd and the 4th moments

  • Skewness (coefficient d’asymétrie): $\frac{E[(X-E[X])^3]}{\sigma^3}$

If Skewness' value is near $0$, then the distribution is symmetric. If the value is $\gt 0$ then the distribution is inclined to the right (resp. to the left).

  • Kurtosis (coefficient d’aplatissement): $\frac{E[(X-E[X])^4]}{\sigma^4}$

If Skewness' value is small, then the distribution is equilibrated. If the value is 3, then we have the normal distribution.


Moments in R

In R, you may use one of these libraries

  • e1071
  • moments