The kthmoment about the mean (or kthcentral moment) of a real-valued random variableX is the quantity E[(X − E[X])k], where E is the expectation operator. Some random variables have no mean, in which case the moment about the mean is not defined. The kth moment about the mean is often denoted μk. For a continuous univariate probability distribution with probability density functionf(x) the moment about the mean μ is
Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the nth_order moment about the origin to the moment about the mean is
where m is the mean of the distribution, and the moment about the origin is given by
The first moment about the mean is zero. The second moment about the mean is called the variance, and is usually denoted σ2, where σ represents the standard deviation. The third and fourth moments about the mean are used to define the standardized moments which are in turn used to define skewness and kurtosis, respectively.
The second centralmoment is the variance, the square root of which is the standard deviation.
The third centralmoment is skewness or the symmetry of the probability distribution.
The centralmoments beyond the third lack this linearity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third centralmoments; the higher cumulants have a more complicated relationship with the centralmoments).
The third and fourth moments about the mean are used to define the standardized moments which are in turn used to define skewness and kurtosis, respectively.
For n ≥ 2, the nth centralmoment is translation-invariant, i.e.
For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth centralmoment; for n ≥ 4, the nth cumulant is an nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-degree polynomial in the first n centralmoments.
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