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In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f (x) = e −1/x 2 can be written as a Laurent series.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, [2] although an earlier version of the result was already mentioned in 1671 by James Gregory. [ 3 ] Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis .
4. The solution is to expand the function z in a second-order Taylor series; the expansion is done around the mean values of the several variables x. (Usually the expansion is done to first order; the second-order terms are needed to find the bias in the mean. Those second-order terms are usually dropped when finding the variance; see below). 5.
For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]
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the sum of two random variables is a random variable; the product of two random variables is a random variable; addition and multiplication of random variables are both commutative; and; there is a notion of conjugation of random variables, satisfying (XY) * = Y * X * and X ** = X for all random variables X,Y and coinciding with complex ...
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