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A probability distribution (concentrated) on a finite or countably infinite set of points of a [[Sampling space|sampling space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330601.png" />. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330602.png" /> be the sample points and let
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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be numbers satisfying the conditions
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{{MSC|60E05}}
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[[Category:Distribution theory]]
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A probability distribution (concentrated) on a finite or countably infinite set of points of a [[Sampling space|sampling space]]  $  \Omega $.
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More exactly, let  $  \omega _ {1} , \omega _ {2} \dots $
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be the sample points and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
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p _ {i}  = p ( \omega _ {i} ) ,\ \
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i= 1 , 2 \dots
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$$
  
Relations (1) and (2) fully define a discrete distribution on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330605.png" />, since the probability measure of any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330606.png" /> is defined by the equation
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be numbers satisfying the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330607.png" /></td> </tr></table>
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$$ \tag{2 }
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p _ {i}  \geq  0 ,\  \sum _ { i } p _ {i}  = 1 .
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$$
  
Accordingly, the distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330608.png" /> is said to be discrete if it assumes, with probability one, a finite or a countably infinite number of distinct values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330609.png" /> with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306010.png" />. In the case of a distribution on the real line, the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306011.png" /> has jumps at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306012.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306013.png" />, and is constant in the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306014.png" />. The following discrete distributions occur most frequently: the [[Binomial distribution|binomial distribution]], the [[Geometric distribution|geometric distribution]], the [[Hypergeometric distribution|hypergeometric distribution]], the [[Negative binomial distribution|negative binomial distribution]], the [[Multinomial distribution|multinomial distribution]], and the [[Poisson distribution|Poisson distribution]].
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Relations (1) and (2) fully define a discrete distribution on the space  $  \Omega $,  
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since the probability measure of any set  $  A \subset  \Omega $
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is defined by the equation
  
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$$
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P ( A)  =  \sum _ {\{ {i } : {\omega _ {i} \in A } \}
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} p _ {i} .
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$$
  
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Accordingly, the distribution of a random variable  $  X ( \omega ) $
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is said to be discrete if it assumes, with probability one, a finite or a countably infinite number of distinct values  $  x _ {i} $
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with probabilities  $  p _ {i} = {\mathsf P} \{  \omega  : {X ( \omega ) = x _ {i} } \} $.
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In the case of a distribution on the real line, the distribution function  $  F ( x) = \sum _ {\{ {i }  : {x _ {i} < x } \} } p _ {i} $
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has jumps at the points  $  x _ {i} $
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equal to  $  p _ {i} = F ( x _ {i} + 0 ) - F ( x _ {i} ) $,
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and is constant in the intervals  $  [ x _ {i} , x _ {i+} 1 ) $.
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The following discrete distributions occur most frequently: the [[Binomial distribution|binomial distribution]], the [[Geometric distribution|geometric distribution]], the [[Hypergeometric distribution|hypergeometric distribution]], the [[Negative binomial distribution|negative binomial distribution]], the [[Multinomial distribution|multinomial distribution]], and the [[Poisson distribution|Poisson distribution]].
  
 
====Comments====
 
====Comments====
A word of caution. In the Russian literature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306015.png" />, whereas in Western literature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306016.png" />. So the distribution functions are slightly different: left continuous in the Russian literature, and right continuous in the Western literature.
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A word of caution. In the Russian literature, $  F ( x) = {\mathsf P} \{ X < x \} $,  
 +
whereas in Western literature $  F ( x) = {\mathsf P} \{ X \leq  x \} $.  
 +
So the distribution functions are slightly different: left continuous in the Russian literature, and right continuous in the Western literature.

Latest revision as of 19:35, 5 June 2020


2020 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

A probability distribution (concentrated) on a finite or countably infinite set of points of a sampling space $ \Omega $. More exactly, let $ \omega _ {1} , \omega _ {2} \dots $ be the sample points and let

$$ \tag{1 } p _ {i} = p ( \omega _ {i} ) ,\ \ i= 1 , 2 \dots $$

be numbers satisfying the conditions

$$ \tag{2 } p _ {i} \geq 0 ,\ \sum _ { i } p _ {i} = 1 . $$

Relations (1) and (2) fully define a discrete distribution on the space $ \Omega $, since the probability measure of any set $ A \subset \Omega $ is defined by the equation

$$ P ( A) = \sum _ {\{ {i } : {\omega _ {i} \in A } \} } p _ {i} . $$

Accordingly, the distribution of a random variable $ X ( \omega ) $ is said to be discrete if it assumes, with probability one, a finite or a countably infinite number of distinct values $ x _ {i} $ with probabilities $ p _ {i} = {\mathsf P} \{ \omega : {X ( \omega ) = x _ {i} } \} $. In the case of a distribution on the real line, the distribution function $ F ( x) = \sum _ {\{ {i } : {x _ {i} < x } \} } p _ {i} $ has jumps at the points $ x _ {i} $ equal to $ p _ {i} = F ( x _ {i} + 0 ) - F ( x _ {i} ) $, and is constant in the intervals $ [ x _ {i} , x _ {i+} 1 ) $. The following discrete distributions occur most frequently: the binomial distribution, the geometric distribution, the hypergeometric distribution, the negative binomial distribution, the multinomial distribution, and the Poisson distribution.

Comments

A word of caution. In the Russian literature, $ F ( x) = {\mathsf P} \{ X < x \} $, whereas in Western literature $ F ( x) = {\mathsf P} \{ X \leq x \} $. So the distribution functions are slightly different: left continuous in the Russian literature, and right continuous in the Western literature.

How to Cite This Entry:
Discrete distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_distribution&oldid=16276
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article