Taylor expansions for the moments of functions of random variables
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.Since the second 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,Second moment
Similarly,
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,
The second order approximation, when X follows a normal distribution, is:
