Approximating the expected value and variance of the function of a (continuous univariate) random variable

Question

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:

$$ \begin{align*} E[g(x)] & \simeq g(E[x]) + \frac{\mathrm{Var}[X]}{2}g''(E[X])\\ \mathrm{Var}[g(x)]& \simeq \left( g'(E[X]) \right)^2\mathrm{Var}[X] \, \mathrm{.} \end{align*} $$

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)?

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EDIT

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

Answer 1

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

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

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

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