Rohmoment

Rohmoment, also known as raw moment or moment about the origin, is a basic quantity in probability and statistics. For a random variable X with a well-defined k-th moment, the k-th Rohmoment is m_k = E[X^k], the expected value of X raised to the k-th power. This contrasts with central moments, which are taken about the mean μ = E[X].

To compute m_k, use the appropriate expectation: if X is continuous with density f, m_k = ∫_{-∞}^{∞} x^k f(x) dx; if X is discrete with probabilities p_i at values x_i, m_k = Σ_i p_i x_i^k.

The first Rohmoment m1 equals μ, the mean. The second Rohmoment m2 relates to variance by Var(X) = m2 - μ^2. Higher raw moments describe tail behavior and shape; they can be used to derive central moments via the binomial expansion: μ_k' = E[(X - μ)^k] = Σ_{j=0}^k binomial(k,j) (-μ)^{k-j} m_j.

Example: If X is uniformly distributed on {1,2,3,4,5,6} (a fair die), m1 = 3.5, m2 = (1^2+...+6^2)/6 = 91/6 ≈ 15.17, m3 and m4 can be computed similarly.

Uses and notes: Rohmoments are central to moment generating functions and parameter estimation, and they help summarize distributional properties. However, not all sequences of moments correspond to a distribution; the moment problem (Hamburger, Stieltjes) arises in such contexts, and higher moments can be sensitive to outliers.