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{{MSC|60E05}}

[[Category:Distribution theory]]

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

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

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====
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.