Difference between revisions of "Variance"
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Let $\mathbb{T}$ be a [[time scale]]. Let $X$ be a random variable with [[probability density function]] $f \colon \mathbb{T} \rightarrow \mathbb{R}$. Then the variance of $X$ is defined by the formula | Let $\mathbb{T}$ be a [[time scale]]. Let $X$ be a random variable with [[probability density function]] $f \colon \mathbb{T} \rightarrow \mathbb{R}$. Then the variance of $X$ is defined by the formula | ||
| − | $$\ | + | $$\mathrm{Var}_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$ |
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
| − | $$\ | + | $$\mathrm{Var}_{\mathbb{T}}(X) = \mathrm{E}_{\mathbb{T}}(X^2) - (\mathrm{E}_{\mathbb{T}}(X))^2.$$ |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | =Examples= | ||
| + | [[Variance of uniform distribution]]<br /> | ||
| + | [[Variance of exponential distribution]]<br /> | ||
| + | [[Variance of gamma distribution]]<br /> | ||
| + | |||
| + | =References= | ||
| + | [https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalities by Thomas Matthews] | ||
Latest revision as of 15:58, 22 September 2016
Let $\mathbb{T}$ be a time scale. Let $X$ be a random variable with probability density function $f \colon \mathbb{T} \rightarrow \mathbb{R}$. Then the variance of $X$ is defined by the formula $$\mathrm{Var}_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$
Properties
Theorem: The following formula holds: $$\mathrm{Var}_{\mathbb{T}}(X) = \mathrm{E}_{\mathbb{T}}(X^2) - (\mathrm{E}_{\mathbb{T}}(X))^2.$$
Proof: █
Examples
Variance of uniform distribution
Variance of exponential distribution
Variance of gamma distribution
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
