Momente

Momente, in German mathematical and statistical usage, are quantitative measures that summarize certain aspects of a probability distribution or a data set. They provide a way to capture information about the shape, location, and spread of the distribution.

A common distinction is between raw moments and central moments. The k-th raw moment of a random variable X is defined as μ′k = E[Xk], where E denotes the expected value. The k-th central moment is μk = E[(X − E[X])k], computed with respect to the distribution’s mean. Central moments describe deviations from the mean, while raw moments describe the distribution relative to zero.

Moment generating functions offer a compact way to encode moments. For a random variable X, the moment generating function MX(t) = E[e(tX)] exists for some range of t, and the k-th derivative at t = 0 gives the k-th raw moment: μ′k = MX^(k)(0). Central moments can be obtained from raw moments and the mean.

In practice, the first few moments have direct interpretations: the first raw moment is the mean, the second central moment is the variance, and higher moments provide information about skewness (asymmetry) and kurtosis (peakedness) of the distribution. Standardized moments, such as the third (skewness) and fourth (excess kurtosis) moments, describe deviations from normality.

Applications of moments span statistics, physics, and engineering. They are used to summarize data distributions, estimate parameters, and describe physical systems through concepts like multipole moments. In German-language literature, Momente is the standard term for these measures, with terminology closely aligned to its English counterpart.