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. First moment
Given  and , the mean and the variance of , respectively, a Taylor expansion of the expected value of  can be found via

 

Since  the second term vanishes. Also,  is . Therefore,

. 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 obtained using a second order approximation, following 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. Indeed, we take . With , we get . The variance is then computed using the formula 
. An example is,

The second order approximation, when X follows a normal distribution, is:

First product moment
To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows. First, note that . Since a second-order expansion for  has already been derived above, it only remains to find . Treating  as a two-variable function, the second-order Taylor expansion is as follows:

 

Taking expectation of the above and simplifying—making use of the identities  and —leads to . Hence,

Random vectors
If X is a random vector, the approximations for the mean and variance of  are given by
 
Here  and  denote the gradient and the Hessian matrix respectively, and  is the covariance matrix of X. See also
Propagation of uncertainty
WKB approximation
Delta method

Notes

Further reading

Statistical approximations
Algebra of random variables
Moment (mathematics)