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.
Notice that , the 2nd term disappears. Also is . Therefore,
where and are the mean and variance of X respectively.
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,
The above is using a first order approximation unlike for the method used in estimating the first moment. It will be a poor approximation in cases where is highly non-linear. This is a special case of the delta method. For example,