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''probability distribution, probability''

{{MSC|60-01}}

A real non-negative function  $  {\mathsf P} $
on a class  $  {\mathcal A} $
of subsets (events) of a non-empty set  $  \Omega $
(the space of elementary events) forming a  $  \sigma $-
field (i.e. a set closed with respect to countable set-theoretic operations) such that

$$ 
{\mathsf P} ( \Omega )  =  1 \  \textrm{ and } \ \ 
{\mathsf P} \left ( \cup _ { i=1} ^  \infty   A _ {i} \right )  = \ 
\sum _ { i=1 } ^  \infty   {\mathsf P} ( A _ {i} )
$$

if  $  A _ {i} \cap A _ {j} = \emptyset $
for  $  i \neq j $(
$  \sigma $-
additivity).

===Examples of probability measures.===

1)  $  \Omega = \{ 1, 2 \} $; 
$  {\mathcal A} $
is the class of all subsets of  $  \Omega $; 
$  {\mathsf P} ( \{ 1 \} ) = {\mathsf P} ( \{ 2 \} ) = 1 / 2 $
(this probability measure corresponds to a random experiment consisting in throwing a symmetrical coin; if heads correspond to 1 while tails correspond to 2, the probability of throwing heads (tails) is 1/2);

2)  $  \Omega = \{ 0, 1 , .  .  . \} $; 
$  {\mathcal A} $
is the class of all subsets of  $  \Omega $;

$$ 
{\mathsf P} ( \{ k \} )  = \ 

\frac{\lambda  ^ {k} }{k!}
 e ^ {- \lambda } ,
$$

where  $  \lambda > 0 $
(the [[Poisson distribution|Poisson distribution]]);

3)  $  \Omega = \mathbf R  ^ {1} $; 
$  {\mathcal A} $
is the class of Borel subsets of  $  \mathbf R  ^ {1} $;

$$ 
{\mathsf P} ( A)  = \ 

\frac{1}{\sqrt {2 \pi } }
 \int\limits _ { A } e ^ {- x  ^ {2} /2 }  dx
$$
(the [[Normal distribution|normal distribution]]);

4)  $  \Omega = C _ {0} [ 0, 1] $
is the space of continuous real functions  $  x( t) $
on  $  [ 0, 1] $
that vanish at the point zero;  $  {\mathcal A} $
is the class of Borel subsets with respect to the topology of uniform convergence;  $  {\mathsf P} $
is the measure which is uniquely defined by the formula

$$ 
{\mathsf P} ( x : a _ {i} < x ( t _ {i} ) < b _ {i} ,\ 
i = 1 \dots n) =
$$

$$ 
= \ 
( 2 \pi )  ^ {- n/2} \prod _ { i=1 } ^ { n }  ( t _ {i} - t _ {i-1} )  ^ {- 1/2} \times
$$

$$ 
\times 
\int\limits _ { a _ {1} } ^ { {b _ 1 } } \dots \int\limits
_ {a _ {n} } ^ { {b _ n} }  \mathop{\rm exp} \left \{ - 
\frac{1}{2}

\sum _ { i=1 } ^ { n }  
\frac{( x _ {i} - x _ {i-1} )  ^ {2} }{t _ {i} - t _ {i-1} }
 \right \}  dx _ {1} \dots d x _ {n} ,
$$

where  $  n $
is an arbitrary natural number and  $  0 = t _ {0} < t _ {1} < \dots < t _ {n} \leq  1 $
(the [[Wiener measure|Wiener measure]]).

====References====
{|
|valign="top"|{{Ref|K}}|| A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) {{MR|0032961}} {{ZBL|}}
|-
|valign="top"|{{Ref|G}}|| B.V. Gnedenko,   [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962) (Translated from Russian)
|}

====Comments====

====References====
{|
|valign="top"|{{Ref|B}}|| P. Billingsley, "Probability and measure" , Wiley (1979) {{MR|0534323}} {{ZBL|0411.60001}}
|}