Difference between revisions of "Variance"
From timescalewiki
(→Examples) |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 12: | Line 12: | ||
=Examples= | =Examples= | ||
| − | + | [[Variance of uniform distribution]]<br /> | |
| − | + | [[Variance of exponential distribution]]<br /> | |
| + | [[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
