Probability MomentsExp - Revision history

Rb at 21:24, 29 August 2013

2013-08-29T21:24:19Z

2013-08-29T21:20:46Z

2013-08-29T21:14:43Z

2013-08-15T15:27:26Z

‎Uniform Distribution

2013-08-15T15:27:15Z

‎Uniform Distribution

2013-08-15T15:26:56Z

‎Uniform Distribution

2013-08-15T15:26:36Z

‎Uniform Distribution

2013-08-15T15:26:21Z

‎Uniform Distribution

2013-08-15T15:25:56Z

‎Uniform Distribution

2013-08-15T15:25:26Z

‎Uniform Distribution

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 === Additional resources ===
-* The list above is a very small, but useful, selection of continuous distributions that have been proposed. If your hunger for continuous distributions is not yet satisfied check out this list on [http://mathworld.wolfram.com/topics/ContinuousDistributions.html MathWorld].
+* The list above is a very small, but useful, selection of continuous distributions that have been proposed. If your hunger for continuous distributions is not yet satisfied check out this list on [http://mathworld.wolfram.com/topics/ContinuousDistributions.html MathWorld]. One of the weirdest distributions is the [http://mathworld.wolfram.com/CauchyDistribution.html Cauchy] distributions. It looks rather innocent but has no moments. Not for the feint-hearted!
 = Footnotes =

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 E\left[ X\right] &=&\lambda \\
 var\left[ X\right] &=&\lambda\end{aligned}</math>
 = Means and Variances for some Continuous Distributions =
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 and this is the density of a <math>\chi _{1}^{2}</math> random variable, noting that <math>\Gamma \left( 1/2\right) =\sqrt{\pi }</math>.
 = Footnotes =

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 Let <math>X</math> be a discrete random variable, which when applied can yield any one of <math>r</math> possible values denoted <math>a_{1},a_{2},...,a_{r}</math>. Suppose that a suitable experiment is performed which can give rise to <math>n</math> observed values of this random variable. Let <math>f_{1}</math> denote the number of times <math>a_{1}</math> occurs, <math>f_{2}</math> the number of times <math>a_{2}</math> occurs, etc, once the experiment has been performed, with <math>\sum_{j=1}^{r}f_{j}=n</math>.
 On the basis of observed sample data the sample mean would be
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 The theoretical mean, <math>E\left[ X\right] </math>, is referred to as the first moment about the origin; other moments are also of interest as follows.
 == Higher order moments ==
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 <math>\begin{aligned}
-f\left( x\right)&=&\frac{1}{b-a} , \quad a < x < b\\
+f\left( x\right) &=&\frac{1}{b-a}, \quad a < x < b \\
-E\left[ X\right] &=&\frac{b+a}{2}\\
+E\left[ X\right] &=&\frac{b+a}{2} \\
-var\left[ X\right] &=&\frac{1}{12}\left( b-a\right) ^{2}
+var\left[ X\right] &=&\frac{1}{12}\left( b-a\right) ^{2}\end{aligned}</math>
 === Exponential distribution ===

← Older revision		Revision as of 15:27, 15 August 2013
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	=== Uniform Distribution ===		=== Uniform Distribution ===
−
−	~~<math>\begin{aligned}~~
−	~~p\left( x\right) &=&\Pr \left( X=x\right) =\frac{\lambda ^{x}}{x!}\exp \left( -\lambda \right) ,\quad x=0,1,2,...;\quad \lambda >0 \\~~
−	~~E\left[ X\right] &=&\lambda \\~~
−	~~var\left[ X\right] &=&\lambda\end{aligned}</math>~~

	<math>\begin{aligned}		<math>\begin{aligned}

@@ Line 278: / Line 278: @@
 <math>\begin{aligned}
 f\left( x\right)&=&\frac{1}{b-a} , \quad a < x < b\\
-E\left[ X\right] &=&\frac{b+a}{2}
+E\left[ X\right] &=&\frac{b+a}{2}\\
 \end{aligned}</math>