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| | Not much to say about myself at all.<br>Hurrey Im here and a member of wmflabs.org.<br>I really hope I'm useful in some way .<br><br>Here is my blog :: [http://safedietplansforwomen.com/waist-to-height-ratio waist height ratio] |
| <!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion
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| of standards used for probability distribution articles such as this one. -->
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| {{Probability distribution|
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| name =Degenerate|
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| type =mass|
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| 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>|
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| 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>|
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| parameters =<math>k_0 \in (-\infty,\infty)\,</math>|
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| support =<math>k=k_0\,</math>|
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| pdf =<math>
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| \begin{matrix}
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| 1 & \mbox{for }k=k_0 \\0 & \mbox{otherwise }
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| \end{matrix}
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| </math>|
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| cdf =<math>
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| \begin{matrix}
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| 0 & \mbox{for }k<k_0 \\1 & \mbox{for }k\ge k_0
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| \end{matrix}
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| </math>|
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| mean =<math>k_0\,</math>|
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| median =<math>k_0\,</math>|
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| mode =<math>k_0\,</math>|
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| variance =<math>0\,</math>|
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| skewness =[[0/0|undefined]]|
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| kurtosis =[[0/0|undefined]]|
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| entropy =<math>0\,</math>|
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| mgf =<math>e^{k_0t}\,</math>|
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| char =<math>e^{ik_0t}\,</math>|
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| }}
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| 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:
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| <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>
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| The [[cumulative distribution function]] of the degenerate distribution is then:
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| <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>
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| ==Constant random variable==
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| 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.
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| Let ''X'': Ω → '''R''' be a random variable defined on a probability space (Ω, ''P''). Then ''X'' is an ''almost surely constant random variable'' if there exists <math> c \in \mathbb{R} </math> such that
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| :<math>\Pr(X = c) = 1,</math>
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| and is furthermore a ''constant random variable'' if
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| :<math>X(\omega) = c, \quad \forall\omega \in \Omega.</math>
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| Note that a constant random variable is almost surely constant, but not necessarily ''vice versa'', since if ''X'' is almost surely constant then there may exist γ ∈ Ω such that ''X''(γ) ≠ ''c'' (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ c) = 0).
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| For practical purposes, the distinction between ''X'' being constant or almost surely constant is unimportant, since the [[probability mass function]] ''f''(''x'') and [[cumulative distribution function]] ''F''(''x'') of ''X'' do not depend on whether ''X'' is constant or 'merely' almost surely constant. In either case,
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| :<math>f(x) = \begin{cases}1, &x = c,\\0, &x \neq c.\end{cases}</math>
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| and
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| :<math>F(x) = \begin{cases}1, &x \geq c,\\0, &x < c.\end{cases}</math>
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| The function ''F''(''x'') is a [[step function]]; in particular it is a [[translation (geometry)|translation]] of the [[Heaviside step function]].
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| ==See also==
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| * [[Dirac delta function]]
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| {{ProbDistributions|miscellaneous}}
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| {{DEFAULTSORT:Degenerate Distribution}}
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| [[Category:Discrete distributions]]
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| [[Category:Types of probability distributions]]
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| [[Category:Infinitely divisible probability distributions]]
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| [[Category:Probability distributions]]
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Not much to say about myself at all.
Hurrey Im here and a member of wmflabs.org.
I really hope I'm useful in some way .
Here is my blog :: waist height ratio