Gross enrolment ratio

2013-12-08T06:31:01Z

117.216.176.13: /* Example */

{{Unreferenced|date=December 2009}}

In [[probability theory]], it is possible to approximate the [[moment (mathematics)|moments]] of a function ''f'' of a [[random variable]] ''X'' using [[Taylor expansion]]s, provided that ''f'' is sufficiently differentiable and that the moments of ''X'' are finite. This technique is often used by [[statistics|statisticians]].


==First moment==
: <math>
\begin{align}
\operatorname{E}\left[f(X)\right] & {} = \operatorname{E}\left[f(\mu_X + \left(X - \mu_X\right))\right] \\
& {} \approx \operatorname{E}\left[f(\mu_X) + f'(\mu_X)\left(X-\mu_X\right) + \frac{1}{2}f''(\mu_X) \left(X - \mu_X\right)^2 \right].
\end{align}
</math>

Noting that <math>E[X-\mu_X]=0</math>, the 2nd term disappears. Also <math>E[(X-\mu_X)^2]</math> is <math>\sigma_X^2</math>. Therefore,

:<math>\operatorname{E}\left[f(X)\right]\approx f(\mu_X) +\frac{f''(\mu_X)}{2}\sigma_X^2</math>
where <math>\mu_X</math> and <math>\sigma^2_X</math> are the mean and variance of X respectively.

It is possible to generalize this to functions of more than one variable using [[Taylor expansion#Taylor series in several variables|multivariate Taylor expansions]]. For example,

:<math>\operatorname{E}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]} -\frac{\operatorname{cov}\left[X,Y\right]}{\operatorname{E}\left[Y\right]^2}+\frac{\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{var}\left[Y\right]</math>

==Second moment==
Analogously,

:<math>\operatorname{var}\left[f(X)\right]\approx \left(f'(\operatorname{E}\left[X\right])\right)^2\operatorname{var}\left[X\right] = \left(f'(\mu_X)\right)^2\sigma^2_X.</math>

The above is using a first order approximation unlike for the method used in estimating the first moment. It will be a poor approximation in cases where <math>f(X)</math> is highly non-linear. This is a special case of the [[delta method]]. For example,

:<math>\operatorname{var}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{var}\left[X\right]}{\operatorname{E}\left[Y\right]^2}-\frac{2\operatorname{E}\left[X\right]}{\operatorname{E}\left[Y\right]^3}\operatorname{cov}\left[X,Y\right]+\frac{\operatorname{E}\left[X\right]^2}{\operatorname{E}\left[Y\right]^4}\operatorname{var}\left[Y\right].</math>

==See also==
*[[Propagation of uncertainty]]
*[[WKB approximation]]
*http://www.stanford.edu/class/cme308/notes/TaylorAppDeltaMethod.pdf

{{DEFAULTSORT:Taylor Expansions For The Moments Of Functions Of Random Variables}}
[[Category:Statistical approximations]]
[[Category:Algebra of random variables]]

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Gross enrolment ratio