secondmoment

Second moment is a term used in mathematics, statistics, and related fields to denote the expected value of the square of a random variable. Formally, if X is a real-valued random variable with finite second moment, the second moment about the origin is E[X^2]. The second moment about the mean, or the second central moment, is Var(X) = E[(X − E[X])^2] = E[X^2] − (E[X])^2. These two moments are related by E[X^2] = Var(X) + (E[X])^2.

For discrete random variables, E[X^2] = sum_i x_i^2 p_i, where p_i are the probabilities of values x_i. For continuous variables, E[X^2] = ∫ x^2 f(x) dx, provided the integral converges.

The second moment plays a central role in measuring the scale of a random variable in the L^2 sense, since E[X^2] = ||X||_2^2 for square-integrable X. It also underpins the computation of variance and standard deviation, and features in moment-based estimation and the method of moments. In probability theory, many results and inequalities, such as Chebyshev’s inequality, are formulated in terms of the second central moment.

In many distributions, explicit formulas exist: for a normal distribution N(μ, σ^2), E[X^2] = σ^2 + μ^2. The term “second moment” also appears in applied contexts beyond probability, notably as the second moment of area (area moment of inertia), used in structural engineering to assess bending stiffness, defined by ∫∫ y^2 dA over a region.