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In mathematics, the trigonometric moment problem is formulated as follows: given a sequence {}, does there exist a distribution function on the interval [,] such that: [1] [2] = (), with = ¯ for . In case the sequence is finite, i.e., { c k } k = 0 n < ∞ {\displaystyle \{c_{k}\}_{k=0}^{n<\infty }} , it is referred to as the truncated ...
The p-th central moment of a measure μ on the measurable space (M, B(M)) about a given point x 0 ∈ M is defined to be (,) (). μ is said to have finite p-th central moment if the p-th central moment of μ about x 0 is finite for some x 0 ∈ M.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a , b , and c are the lengths of the three sides of the triangle, and α , β , and γ are the angles opposite those three respective sides.
In econometrics, the method of simulated moments (MSM) (also called simulated method of moments [1]) is a structural estimation technique introduced by Daniel McFadden. [2] It extends the generalized method of moments to cases where theoretical moment functions cannot be evaluated directly, such as when moment functions involve high-dimensional integrals.
In the case m 0 = 1, this is equivalent to the existence of a random variable X supported on [0, 1], such that E[X n] = m n. The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line [0, ∞), and in the Hamburger moment ...
A related functional that shares the translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κ n (X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ...
In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences. [1] Suppose X is a random variable and that all of the moments exist.