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| | Hi there. Let me start by introducing the writer, her title is Myrtle Cleary. Managing individuals has been his day occupation for a while. To collect cash is what his family members and him appreciate. Years ago he moved to North Dakota and his family members enjoys it.<br><br>Here is my homepage - [http://www.cam4teens.com/blog/84472 at home 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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| |<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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Hi there. Let me start by introducing the writer, her title is Myrtle Cleary. Managing individuals has been his day occupation for a while. To collect cash is what his family members and him appreciate. Years ago he moved to North Dakota and his family members enjoys it.
Here is my homepage - at home std test