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''probability distribution, probability''
 
''probability distribution, probability''
  
 
{{MSC|60-01}}
 
{{MSC|60-01}}
  
A real non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749301.png" /> on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749302.png" /> of subsets (events) of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749303.png" /> (the space of elementary events) forming a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749305.png" />-field (i.e. a set closed with respect to countable set-theoretic operations) such that
+
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
  
<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/p/p074/p074930/p0749306.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} ( \Omega )  = 1 \  \textrm{ and } \ \
 +
{\mathsf P} \left ( \cup _ { i=1} ^  \infty  A _ {i} \right )  = \
 +
\sum _ { i=1 } ^  \infty  {\mathsf P} ( A _ {i} )
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749307.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749308.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749309.png" />-additivity).
+
if $  A _ {i} \cap A _ {j} = \emptyset $
 +
for $  i \neq j $(
 +
$  \sigma $-
 +
additivity).
  
 
===Examples of probability measures.===
 
===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);
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493010.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493011.png" /> is the class of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493013.png" /> (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 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493014.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493015.png" /> is the class of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493016.png" />;
+
$$
 +
{\mathsf P} ( \{ k \} ) = \
  
<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/p/p074/p074930/p07493017.png" /></td> </tr></table>
+
\frac{\lambda  ^ {k} }{k!}
 +
e ^ {- \lambda } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493018.png" /> (the [[Poisson distribution|Poisson distribution]]);
+
where $  \lambda > 0 $
 +
(the [[Poisson distribution|Poisson distribution]]);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493019.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493020.png" /> is the class of Borel subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493021.png" />;
+
3) $  \Omega = \mathbf R  ^ {1} $;  
 +
$  {\mathcal A} $
 +
is the class of Borel subsets of $  \mathbf R  ^ {1} $;
  
<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/p/p074/p074930/p07493022.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} ( A)  = \
  
 +
\frac{1}{\sqrt {2 \pi } }
 +
\int\limits _ { A } e ^ {- x  ^ {2} /2 }  dx
 +
$$
 
(the [[Normal distribution|normal distribution]]);
 
(the [[Normal distribution|normal distribution]]);
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493023.png" /> is the space of continuous real functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493025.png" /> that vanish at the point zero; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493026.png" /> is the class of Borel subsets with respect to the topology of uniform convergence; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493027.png" /> is the measure which is uniquely defined by the formula
+
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
 +
 
 +
$$
 +
{\mathsf P} ( x : a _ {i} < x ( t _ {i} ) < b _ {i} ,\
 +
i = 1 \dots n) =
 +
$$
  
<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/p/p074/p074930/p07493028.png" /></td> </tr></table>
+
$$
 +
= \
 +
( 2 \pi )  ^ {- n/2} \prod _ { i=1 } ^ { n }  ( t _ {i} - t _ {i-1} )  ^ {- 1/2} \times
 +
$$
  
<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/p/p074/p074930/p07493029.png" /></td> </tr></table>
+
$$
 +
\times
 +
\int\limits _ { a _ {1} } ^ { {b _ 1 } } \dots \int\limits
 +
_ {a _ {n} } ^ { {b _ n} }  \mathop{\rm exp} \left \{ -  
 +
\frac{1}{2}
  
<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/p/p074/p074930/p07493030.png" /></td> </tr></table>
+
\sum _ { i=1 } ^ { n } 
 +
\frac{( x _ {i} - x _ {i-1} )  ^ {2} }{t _ {i} - t _ {i-1} }
 +
\right \}  dx _ {1} \dots d x _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493031.png" /> is an arbitrary natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493032.png" /> (the [[Wiener measure|Wiener measure]]).
+
where $  n $
 +
is an arbitrary natural number and $  0 = t _ {0} < t _ {1} < \dots < t _ {n} \leq  1 $
 +
(the [[Wiener measure|Wiener measure]]).
  
 
====References====
 
====References====
Line 44: Line 106:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====

Latest revision as of 19:48, 8 January 2021


probability distribution, probability

2020 Mathematics Subject Classification: Primary: 60-01 [MSN][ZBL]

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);

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);

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).

References

[K] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) MR0032961
[G] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)

Comments

References

[B] P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001
How to Cite This Entry:
Probability measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_measure&oldid=26924
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article