Difference between revisions of "Uniform probability distribution"
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-->\frac{1}{(b-a)+1} & \text{for }c\in\{a,\ldots,b\}\subseteq\mathbb{N}_{\ge 0} \\<!-- | -->\frac{1}{(b-a)+1} & \text{for }c\in\{a,\ldots,b\}\subseteq\mathbb{N}_{\ge 0} \\<!-- | ||
-->0 & \text{otherwise}\end{array}\right.}} | -->0 & \text{otherwise}\end{array}\right.}} | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Fundamental probability distributions navbox|open}} | ||
| + | {{Definition|Statistics|Probability|Elementary Probability}} | ||
| + | {{Probability Distribution|fund=yes}} | ||
Latest revision as of 05:41, 15 January 2018
Contents
Definition
There are a few distinct cases we may define the uniform distribution on, however in any case the concept is clear:
The total probability, [ilmath]1[/ilmath], is spread evenly, or uniformly over the entire sample space, here denoted [ilmath]S[/ilmath], of a probability space here denoted [ilmath](S,\Omega,\mathbb{P})[/ilmath]
Discrete subset of [ilmath]\mathbb{N}_{\ge 0} [/ilmath]
We will cover the common cases, and their notation, first:
- for [ilmath]a,b\in\mathbb{N}_{\ge 0} [/ilmath] we have: [ilmath]X\sim\text{Uni}(a,b)[/ilmath] to mean:
- We form a probability space, [ilmath](S,\Omega,\mathbb{P})[/ilmath]
- [ilmath]S:\eq\{a,a+1,\ldots,b-1,b\}\subseteq\mathbb{N}_{\ge 0} [/ilmath]
- [ilmath]\Omega:\eq[/ilmath][ilmath]\sigma(S)[/ilmath] which in this case means: [ilmath]\Omega\eq[/ilmath][ilmath]\mathcal{P}(S)[/ilmath]
- We form a probability space, [ilmath](S,\Omega,\mathbb{P})[/ilmath]
Snippets
- for [ilmath]c\in\mathbb{R} [/ilmath] we define: [math]\mathbb{P}[X\eq c]:\eq\left\{\begin{array}{lr}\frac{1}{(b-a)+1} & \text{for }c\in\{a,\ldots,b\}\subseteq\mathbb{N}_{\ge 0} \\0 & \text{otherwise}\end{array}\right.[/math]
References
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