Difference between revisions of "Discrete distribution"
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+ | $#C+1 = 16 : ~/encyclopedia/old_files/data/D033/D.0303060 Discrete distribution | ||
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{{MSC|60E05}} | {{MSC|60E05}} | ||
[[Category:Distribution theory]] | [[Category:Distribution theory]] | ||
− | A probability distribution (concentrated) on a finite or countably infinite set of points of a [[Sampling space|sampling space]] | + | A probability distribution (concentrated) on a finite or countably infinite set of points of a [[Sampling space|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 | 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|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, < | + | 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.
Discrete distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_distribution&oldid=46730