Difference between revisions of "Variance"

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=Examples=
 
=Examples=
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[[Variance of uniform distribution]]<br />
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[[Variance of exponential distribution]]<br />
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[[Variance of gamma distribution]]<br />
  
 
=References=
 
=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]
 
[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