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| {{Unreferenced|date=December 2009}}
| | Nice to meet you, my title is Refugia. One of the issues she loves most is to read comics and she'll be starting some thing else along with it. Since she was 18 she's been working as a receptionist but her marketing never arrives. Puerto Rico is where he and his spouse reside.<br><br>Have a look at my website [http://www.streaming.iwarrior.net/user/WMcneil over the counter std test] |
| <!-- ==CDF method== will do this later -->
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| 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]].
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| <!--
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| ::{|
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| |-
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| |<math>\mu</math>
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| |<math> = \operatorname{E}\left[X\right]</math>
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| |-
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| |<math>\sigma^2</math>
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| |<math> = \operatorname{var}\left[X\right]</math>
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| |}-->
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| ==First moment==
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| : <math>
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| \begin{align}
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| \operatorname{E}\left[f(X)\right] & {} = \operatorname{E}\left[f(\mu_X + \left(X - \mu_X\right))\right] \\
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| & {} \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].
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| \end{align}
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| </math>
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| 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,
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| :<math>\operatorname{E}\left[f(X)\right]\approx f(\mu_X) +\frac{f''(\mu_X)}{2}\sigma_X^2</math>
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| where <math>\mu_X</math> and <math>\sigma^2_X</math> are the mean and variance of X respectively. | |
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| 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,
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| :<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>
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| ==Second moment==
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| Analogously,
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| :<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>
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| 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,
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| :<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>
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| ==See also==
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| *[[Propagation of uncertainty]]
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| *[[WKB approximation]]
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| *http://www.stanford.edu/class/cme308/notes/TaylorAppDeltaMethod.pdf
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| {{DEFAULTSORT:Taylor Expansions For The Moments Of Functions Of Random Variables}}
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| [[Category:Statistical approximations]]
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| [[Category:Algebra of random variables]]
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Nice to meet you, my title is Refugia. One of the issues she loves most is to read comics and she'll be starting some thing else along with it. Since she was 18 she's been working as a receptionist but her marketing never arrives. Puerto Rico is where he and his spouse reside.
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