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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.

First moment


Given and , the mean and the variance of , respectively,[1] 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



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:[2]

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[3]

Here and denote the gradient and the Hessian matrix respectively, and is the covariance matrix of X.

See also



  1. ^ a b Haym Benaroya, Seon Mi Han, and Mark Nagurka. Probability Models in Engineering and Science. CRC Press, 2005, p166.
  2. ^ Hendeby, Gustaf; Gustafsson, Fredrik. "ON NONLINEAR TRANSFORMATIONS OF GAUSSIAN DISTRIBUTIONS" (PDF). Retrieved 5 October 2017.
  3. ^ Rego, Bruno V.; Weiss, Dar; Bersi, Matthew R.; Humphrey, Jay D. (14 December 2021). "Uncertainty quantification in subject‐specific estimation of local vessel mechanical properties". International Journal for Numerical Methods in Biomedical Engineering. 37 (12): e3535. doi:10.1002/cnm.3535. ISSN 2040-7939. PMC 9019846. PMID 34605615.

Further reading