Difference between revisions of "Variance"
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| − | [https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1] | + | [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] |
Revision as of 17:25, 23 November 2014
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 $$\mathbb{V}ar_{\mathbb{T}}(X) = \dfrac{d^2 C_f}{dz^2}(0).$$
Properties
Theorem: The following formula holds: $$\mathbb{V}ar_{\mathbb{T}}(X) = \mathbb{E}_{\mathbb{T}}(X^2) - (\mathbb{E}_{\mathbb{T}}(X))^2.$$
Proof: proof goes here █
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
