Grundmomenten

Grundmomenten, also known as raw moments or moments about the origin, are a basic set of numerical characteristics of a random variable X. For integers k ≥ 0, the k-th Grundmoment is defined by m_k = E[X^k], the expected value of X raised to the k-th power. The zeroth moment m_0 equals E[X^0] = 1 for any nondegenerate distribution, and the first moment m_1 is the mean of X. The second moment m_2 is related to the variance via Var(X) = m_2 − m_1^2.

These moments form a sequence that, under suitable conditions, characterizes the distribution of X. Central moments μ_k = E[(X − μ)^k], where μ = m_1, measure deviations from the mean and are connected to raw moments through binomial expansions (for example, μ_2 = Var(X)). Higher-order raw moments capture aspects such as skewness and kurtosis when combined with lower moments.

Generating functions provide a compact representation: the moment generating function M_X(t) = E[e^{tX}] has the series expansion M_X(t) = ∑_{k=0}^∞ m_k t^k / k! in a neighborhood of t = 0, whenever the moments exist. In practice, a finite set of moments is often used, with m_1 giving the mean, m_2 the basis for variance, and higher-order moments contributing to shape descriptors when combined into standardized measures.

Applications include parameter estimation via the method of moments, distribution characterization, and theoretical analyses in probability and statistics. Limitations arise because not all distributions have finite moments (e.g., Cauchy), and a given moment sequence may not uniquely determine a distribution (moment problems such as Hamburger and Stieltjes). Multivariate Grundmomenten extend these concepts to joint distributions.