Dirichlet's theorem on arithmetic progressions: Difference between revisions

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<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion
of standards used for probability distribution articles such as this one. -->
{{Probability distribution|
  name      =Degenerate|
  type      =mass|
  pdf_image  =[[Image:Degenerate distribution PMF.png|325px|Plot of the degenerate distribution PMF for k<sub>0</sub>=0]]<br /><small>PMF for k<sub>0</sub>=0. The horizontal axis is the index ''i'' of ''k<sub>i</sub>''.</small>|
  cdf_image  =[[Image:Degenerate distribution CDF.png|325px|Plot of the degenerate distribution CDF for k<sub>0</sub>=0]]<br /><small>CDF for k<sub>0</sub>=0. The horizontal axis is the index ''i'' of ''k<sub>i</sub>''.</small>|
  parameters =<math>k_0 \in (-\infty,\infty)\,</math>|
  support    =<math>k=k_0\,</math>|
  pdf        =<math>
    \begin{matrix}
    1 & \mbox{for }k=k_0 \\0 & \mbox{otherwise }
    \end{matrix}
    </math>|
  cdf        =<math>
    \begin{matrix}
    0 & \mbox{for }k<k_0 \\1 & \mbox{for }k\ge k_0
    \end{matrix}
    </math>|
  mean      =<math>k_0\,</math>|
  median    =<math>k_0\,</math>|
  mode      =<math>k_0\,</math>|
  variance  =<math>0\,</math>|
  skewness  =[[0/0|undefined]]|
  kurtosis  =[[0/0|undefined]]|
  entropy    =<math>0\,</math>|
  mgf        =<math>e^{k_0t}\,</math>|
  char      =<math>e^{ik_0t}\,</math>|
}}


In [[mathematics]], a '''degenerate distribution''' is the [[probability distribution]] of a [[random variable]] which only takes a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. While this distribution does not appear [[randomness|random]] in the everyday sense of the word, it does satisfy the definition of random variable.
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The degenerate distribution is localized at a point ''k''<sub>0</sub> on the [[real line]]. The probability mass function is given by:
 
<math>f(k;k_0)=\left\{\begin{matrix} 1, & \mbox{if }k=k_0 \\ 0, & \mbox{if }k \ne k_0 \end{matrix}\right.</math>
 
The [[cumulative distribution function]] of the degenerate distribution is then:
 
<math>F(k;k_0)=\left\{\begin{matrix} 1, & \mbox{if }k\ge k_0 \\ 0, & \mbox{if }k<k_0 \end{matrix}\right.</math>
 
==Constant random variable==
In [[probability theory]], a '''constant random variable''' is a [[discrete random variable|discrete]] [[random variable]] that takes a [[Constant function|constant]] value, regardless of any [[event (probability theory)|event]] that occurs.  This is technically different from an '''[[almost surely]] constant random variable''', which may take other values, but only on events with probability zero.  Constant and almost surely constant random variables provide a way to deal with constant values in a probabilistic framework.
 
Let  &nbsp;''X'': Ω → '''R'''&nbsp; be a random variable defined on a probability space &nbsp;(Ω, ''P''). Then &nbsp;''X''&nbsp; is an ''almost surely constant random variable'' if there exists <math> c \in \mathbb{R} </math> such that
:<math>\Pr(X = c) = 1,</math>
and is furthermore a ''constant random variable'' if
:<math>X(\omega) = c, \quad \forall\omega \in \Omega.</math>
 
Note that a constant random variable is almost surely constant, but not necessarily ''vice versa'', since if &nbsp;''X''&nbsp; is almost surely constant then there may exist &nbsp;γ ∈ Ω&nbsp; such that &nbsp;''X''(γ) ≠ ''c''&nbsp; (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ c) = 0).
 
For practical purposes, the distinction between &nbsp;''X''&nbsp; being constant or almost surely constant is unimportant, since the [[probability mass function]] &nbsp;''f''(''x'')&nbsp; and [[cumulative distribution function]] &nbsp;''F''(''x'')&nbsp; of &nbsp;''X''&nbsp; do not depend on whether &nbsp;''X''&nbsp; is constant or 'merely' almost surely constant.  In either case,
:<math>f(x) = \begin{cases}1, &x = c,\\0, &x \neq c.\end{cases}</math>
and
:<math>F(x) = \begin{cases}1, &x \geq c,\\0, &x < c.\end{cases}</math>
The function &nbsp;''F''(''x'')&nbsp; is a [[step function]]; in particular it is a [[translation (geometry)|translation]] of the [[Heaviside step function]].
 
==See also==
* [[Dirac delta function]]
 
{{ProbDistributions|miscellaneous}}
 
{{DEFAULTSORT:Degenerate Distribution}}
[[Category:Discrete distributions]]
[[Category:Types of probability distributions]]
[[Category:Infinitely divisible probability distributions]]
[[Category:Probability distributions]]

Latest revision as of 18:19, 27 November 2014

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