Difference between revisions of "Variance"
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[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] | ||
Revision as of 22:05, 14 April 2015
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
Proposition: Let $X$ have the uniform distribution on $[a,b] \cap \mathbb{T}$. Then, $$\mathrm{Var}_{\mathbb{T}}(X)=2\dfrac{h_3(\sigma(b),0)-h_3(a,0)}{\sigma(b)-a}-\left( \dfrac{h_2(\sigma(b),a)}{\sigma(b)-a}+a \right)^2.$$
Proof: █
Theorem
If $X$ with a random variable with the exponential distribution with parameter $\lambda$, then, $$\mathrm{Var}_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2},$$ where $\mathrm{Var}$ denotes variance.
Proof
References
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
