Difference between revisions of "Uniform distribution"
From timescalewiki
| Line 8: | Line 8: | ||
{{:Expected value of uniform distribution}} | {{:Expected value of uniform distribution}} | ||
{{:Variance of uniform distribution}} | {{:Variance of uniform distribution}} | ||
| − | |||
| − | |||
=References= | =References= | ||
[http://scholarsmine.mst.edu/doctoral_dissertations/2241/] | [http://scholarsmine.mst.edu/doctoral_dissertations/2241/] | ||
| + | |||
| + | {{:Probability distributions footer}} | ||
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 $$U_{[a,b]}(t) = \left\{ \begin{array}{ll} \dfrac{1}{\sigma(b)-a} &; a \leq t \leq b \\ 0 &; \mathrm{otherwise} \end{array} \right.$$
Contents
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
Probability distributions | ||
| Uniform distribution | Exponential distribution | Gamma distribution |
