Sajun.org
The ''k''
th '''moment about the mean''' (or ''k''
th '''central moment''') of a real-valued
random variable ''X'' 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 ''k''
th moment about the mean is often denoted μ
''k''. For a continuous univariate
probability distribution with
probability density function ''f''(''x'') the moment about the mean μ is
:<math>
\mu_k
= \left\langle ( x - \langle x \rangle )^k \right\rangle
= \int_{-\infty}^{+\infty} (x - \mu)^k f(x)\,dx.
</math>
Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the n
th-order moment about the origin to the moment about the mean is
:<math>
\mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j m^{n-j},
</math>
where
m is the mean of the distribution, and the moment about the origin is given by
:<math>
\mu'_j = \int_{-\infty}^{+\infty} x^n f(x)\,dx.
</math>
The first moment about the mean is zero. The second moment about the mean is called the
variance, and is usually denoted σ
2, 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.
==See also==
moment (mathematics),
cumulant
