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. This technique is often used by statisticians.

First moment

\begin{align} \operatorname{E}\left[f(X)\right] & {} = \operatorname{E}\left[f(\mu_X + \left(X - \mu_X\right))\right] \\ & {} \approx \operatorname{E}\left[f(\mu_X) + f'(\mu_X)\left(X-\mu_X\right) + \frac{1}{2}f''(\mu_X) \left(X - \mu_X\right)^2 \right]. \end{align}

Notice that $E[X-\mu_X]=0$, the 2nd term disappears. Also $E[(X-\mu_X)^2]$ is $\sigma_X^2$. Therefore,

$\operatorname{E}\left[f(X)\right]\approx f(\mu_X) +\frac{f''(\mu_X)}{2}\sigma_X^2$

where $\mu_X$ and $\sigma^2_X$ 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,

$\operatorname{E}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]} -\frac{\operatorname{cov}\left[X,Y\right]}{\operatorname{E}\left[Y\right]^2}+\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{var}\left[Y\right]$

Second moment

Analogously,

$\operatorname{var}\left[f(X)\right]\approx \left(f'(\operatorname{E}\left[X\right])\right)^2\operatorname{var}\left[X\right] = \left(f'(\mu_X)\right)^2\sigma^2_X.$

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 $f(X)$ is highly non-linear. This is a special case of the delta method. For example,

$\operatorname{var}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{var}\left[X\right]}{\operatorname{E}\left[Y\right]^2}-\frac{2\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{cov}\left[X,Y\right]+\frac{\operatorname{E}\left[X\right]^2}{\operatorname{E}\left[Y\right]^4}\operatorname{var}\left[Y\right].$