# 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:

\$\$
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

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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