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

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:

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{.}

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)?


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

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

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

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