Let $X$ be a univariate continuous random variable (r.v.). Let $g$ be a smooth real function defined on the sample space of $X$.

I have been told that the following approximations are true:

$$

begin{align*}

E[g(x)] & simeq g(E[x]) + frac{mathrm{Var}[X]}{2}g''(E[X])\

mathrm{Var}[g(x)]& simeq left( g'(E[X]) right)^2mathrm{Var}[X] , mathrm{.}

end{align*}

$$

First, is that right?

If so, where could I find a reference for those approximations?

If not, is there a way to accurately approximate $E[g(x)]$ and $mathrm{Var}[g(x)]$ when they are too difficult to be calculated in an exact form (i.e., using integrals)?

EDIT

I have found out that these approximations have to do with Taylor expansions, if I am not wrong.

**Contents**hide

#### Best Answer

The *second* order Taylor approximation around $X= E(X)$ is

$$E[g(X)] simeq EBig [g(E[X]) + g'(E[X])cdot (X-E(X)) + frac 12 g''(E[X])cdot (X-E(X))^2 Big ]$$

The first term is a constant, the expected value of the second term is zero, so we arrive at

$$E[g(X)] simeq g(E[X]) + frac 12 g''(E[X])cdot E[X-E(X)]^2 $$

the last term being the variance.

The *first* order Taylor approximation of $g(X)$ (always around $X= E(X)$) is just $g(E[X]) + g'(E[X])cdot (X-E(X))$ so

$$text{Var}[g(X)] approx text{Var}Big [g(E[X]) + g'(E[X])cdot (X-E(X))Big]$$

$$ = text{Var}Big [g(E[X]) + g'(E[X])cdot X – g'(E[X])cdot E(X) Big]$$

Constant terms have zero variance, and the first and third term are constants. So

$$text{Var}[g(X)] approx text{Var}Big [g'(E[X])cdot X Big] = left( g'(E[X]) right)^2mathrm{Var}[X]$$

### Similar Posts:

- Solved – first and second order approximations of mean and variance
- Solved – Unbiased estimator of exponential of measure of a set
- Solved – Correlation coefficient for a uniform distribution on an ellipse
- Solved – Expectation of truncated normal
- Solved – Should I use a binomial cdf or a normal cdf when flipping coins