Probability measure

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)

References

[B]	P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001