Taylor expansions for the moments of functions of random variables

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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. This technique is often used by statisticians.

First moment


It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,

Second moment


This is a special case of the delta method. For example,