Difference between revisions of "Uniform distribution"

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=References=
 
=References=
 
[http://scholarsmine.mst.edu/doctoral_dissertations/2241/]
 
[http://scholarsmine.mst.edu/doctoral_dissertations/2241/]
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Revision as of 21:53, 14 April 2015

Let $\mathbb{T}$ be a time scale. Let $a,b \in \mathbb{T}$. The uniform distribution on the interval $[a,b] \cap \mathbb{T}$ is given by the formula $$U_{[a,b]}(t) = \left\{ \begin{array}{ll} \dfrac{1}{\sigma(b)-a} &; a \leq t \leq b \\ 0 &; \mathrm{otherwise} \end{array} \right.$$

Properties

Theorem

Let $X$ have the uniform distribution on $[a,b] \cap \mathbb{T}$. Then, $$\mathrm{E}_{\mathbb{T}}(X) = \dfrac{h_2(\sigma(b),a)}{\sigma(b)-a}+a,$$ where $h_2$ denotes the delta hk and $\sigma$ denotes the forward jump.

Proof

References

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:

References

[1]

Probability distributions

Uniform distributionExponential distributionGamma distribution