Editing Algebra of random variables (section)

== Approximations by Taylor series expansions of moments ==
If the [[Moment (mathematics)|moments]] of a certain random variable <math>X</math> are known (or can be determined by integration if the [[probability density function]] is known), then it is possible to approximate the expected value of any general non-linear function <math>f(X)</math> as a [[Taylor expansions for the moments of functions of random variables|Taylor series expansion of the moments]], as follows:

<math display="block">f(X) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{d^n f}{dX^n}\right)_{X=\mu} {\left(X - \mu\right)}^n,</math>
where <math>\mu = \operatorname{E}[X]</math> is the mean value of <math>X</math>.

<math display="block">\begin{align}
\operatorname{E}[f(X)] &= \operatorname{E}\left[ \sum_{n=0}^\infty \frac{1}{n!}\left({d^nf \over dX^n}\right)_{X=\mu} {\left(X-\mu\right)}^n\right] \\
&= \sum_{n=0}^\infty \frac{1}{n!}\left(\frac{d^n f}{dX^n}\right)_{X=\mu} \operatorname{E}\left[{\left(X - \mu\right)}^n\right] \\
&= \sum_{n=0}^\infty \frac{1}{n!}\left({d^nf \over dX^n}\right)_{X=\mu}\mu_n(X),
\end{align}</math>
where <math>\mu_n(X) = \operatorname{E}[(X-\mu)^n]</math> is the ''n''-th moment of <math>X</math> about its mean. Note that by their definition, <math>\mu_0(X)=1</math> and <math>\mu_1(X)=0</math>. The first order term always vanishes but was kept to obtain a closed form expression.

Then,

<math display="block">\operatorname{E}[f(X)] \approx \sum_{n=0}^{n_{\max}} \frac{1}{n!} \left(\frac{d^nf}{dX^n}\right)_{X=\mu}\mu_n(X), </math>
where the Taylor expansion is truncated after the <math>n_{\max} </math>-th moment.

Particularly for functions of [[normal random variable]]s, it is possible to obtain a Taylor expansion in terms of the [[standard normal distribution]]:<ref>{{Cite journal|last=Hernandez|first=Hugo|date=2016|title=Modelling the effect of fluctuation in nonlinear systems using variance algebra - Application to light scattering of ideal gases|journal=ForsChem Research Reports|language=en|volume=2016-1|doi=10.13140/rg.2.2.36501.52969}}</ref>

<math display="block">f(X) = \sum_{n=0}^\infty \frac{\sigma^n}{n!} \left(\frac{d^n f}{dX^n}\right)_{X=\mu} \mu_n(Z),</math>where <math>X \sim N(\mu,\sigma ^2)</math> is a normal random variable, and  <math>Z\sim N(0,1)</math> is the standard normal distribution. Thus,

<math display="block">\operatorname{E}[f(X)]\approx \sum_{n=0}^{n_{\max}} {\sigma ^n \over n!} \left({d^nf \over dX^n}\right)_{X=\mu} \mu_n(Z) , </math> where the moments of the standard normal distribution are given by:

<math display="block">\mu_n(Z) = \begin{cases}
\prod_{i=1}^{n/2}(2i-1), & \text{if } n \text{ is even} \\
0, & \text{if }n\text{ is odd}
\end{cases}</math>

Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as:

<math display="block">\operatorname{Var}[f(X)] \approx \sum_{n=1}^{n_{\max}} \left({\sigma^n \over n!} \left({d^nf \over dX^n}\right)_{X=\mu}\right)^2 \operatorname{Var}[Z^n] + \sum_{n=1}^{n_{\max}} \sum_{m \neq n} \frac{\sigma^{n+m}}{{n!m!}} \left({d^nf \over dX^n}\right)_{X=\mu} \left({d^mf \over dX^m}\right)_{X=\mu} \operatorname{Cov}[Z^n,Z^m],</math>
where
<math display="block">\operatorname{Var}[Z^n] = \begin{cases}
\prod_{i=1}^{n}(2i-1) -\prod_{i=1}^{n/2}(2i-1)^2, & \text{if }n\text{ is even} \\
\prod_{i=1}^{n}(2i-1), & \text{if }n\text{ is odd},
\end{cases}</math>
and
<math display="block">\operatorname{Cov}[Z^n,Z^m] = \begin{cases}
\prod_{i=1}^{(n+m)/2}(2i-1) -\prod_{i=1}^{n/2}(2i-1) \prod_{j=1}^{m/2}(2j-1), & \text{if }n\text{ and }m \text{ are even} \\
\prod_{i=1}^{(n+m)/2}(2i-1), & \text{if }n\text{ and }m\text{ are odd} \\
0, & \text{otherwise}
\end{cases}</math>