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Probability Spaces

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Algebra

Let &Omega. be some abstract set, and let @A_0 be a collection of subsets of &Omega..

Then @A_0 is an #~{algebra} if

  1. &empty. &in. @A_0; _ &Omega. &in. @A_0
  2. if ~A &in. @A_0 then ~A^c &in. @A_0
  3. if ~A_1 ... ~{A_n} &in. @A_0 then &intersect._{~i} ~{A_i} &in. @A_0 ( ~{finite intersection} )
    and &union._{~i} ~{A_i} &in. @A_0 ( ~{finite union} )

Sigma-algebra

A collection , @A, of subsets of &Omega. is a #~{&sigma.-algebra} if.

  1. &empty. &in. @A; _ &Omega. &in. @A
  2. if ~A &in. @A then ~A^c &in. @A
  3. if ~{A_i} &in. @A, {~i} &in. &naturals., then &intersect._{~i} ~{A_i} &in. @A ( ~{countable intersection} )

Note: iii ) &imply. &union._{~i} ~{A_i} &in. @A _ since _ ~{A_i} &in. @A _ &imply. _ ~{A_i}^c &in. @A  _ &imply. _ &intersect._{~i} ~{A_i}^c = ( &union._{~i} ~{A_i} )^c &in. @A _ &imply. _ &union._{~i} ~{A_i} &in. @A

Sigma-algebra generated by subsets

&Omega. some abstract set, and ~S a set of subsets of &Omega., _ then the #~{smallest &sigma.-algebra containing} ~S, or the &sigma.-algebra #~{generated} by ~S is denoted &sigma.( ~S ).

This always exists, as ~S &subseteq. 2^{&Omega.}, and if @F and @G are two sub-&sigma.-algebras containing ~S, then &sigma.( S ) &subseteq. @F &intersect. @G

Borel Sigma-algebra

If V is some topological space, then the #~{Borel &sigma.-algebra} on V, is the &sigma.-algebra generated by all the open sets of V, and is denoted @B( V ).

If &reals. is the real line, then the Borel &sigma.-algebra on &reals., _ @B( &reals. ) _ , is often denoted just _ @B.

Measurable Space

If &Omega. is an abstract set with a &sigma.-algebra @A defined on it, then the pair, ( &Omega., @A ) is called a #~{measurable space}.

Measure

A #~{measure}, &mu., on ( &Omega., @A ) is a set function _ &mu.#: @A -> [ 0, &infty. ) such that
  1. &mu.( &empty. ) = 0
  2. if ~{A_i} &in. @A, {~i} &in. &naturals., and ~{A_i} &intersect. ~{A_j} = &empty. when ~i &neq. ~j
    then &mu.( &union._{~i} ~{A_i} ) = &sum._{~i} &mu.( ~{A_i} ). ( #~{countable additivity} )

Probability Measure

( &Omega., @A ) is a {measurable space}, a measure P: @A &rightarrow. [0,1] is a #~{probability measure} if _ P( &Omega. ) = 1
( &Omega., @A, P ) is called a #~{probability triple} or just a #~{probability space} .

The following results follow directly from the definition:

  1. P( &empty. ) = 0; _ P( &Omega. ) = 1
  2. If ~A and ~B are disjoint then _ P( ~A &union. ~B ) = P( ~A ) + P( ~B )
  3. P( ~A^c ) = 1 &minus. P( ~A )
    [&Omega. = ~A &union. ~A^c _ ( disjoint ), _ so P( ~A ) + P( ~A^c ) = P( &Omega. ) = 1]
  4. ~A &subseteq. ~B &imply. P( ~A ) &le. P( ~B )
  5. P( ~A &union. ~B ) = P( ~A ) + P( ~B ) - P( ~A &intersect. ~B )

Events

If _ ( &Omega., @A, P ) _ is probability space, then elements of @A are known as #~{events} , _ and &Omega. is called the #~{event space}.

Truth Sets

Let @s be a statement which is either true or false for each element of &Omega.. _ The #~{truth set} of @s is simply

#T( @s ) _ = _ \{ &omega. | @s is true for &omega. \}

then note that, if @r is another such statement:

#T( @s ) &intersect. #T( @r ) _ = _ \{ &omega. | @s and @r are true for &omega. \}

#T( @s ) &union. #T( @r ) _ = _ \{ &omega. | @s or @r is true for &omega. \}

( #T( @s ) )^c _ = _ \{ &omega. | @s is not true for &omega. \}

The following nomeclature is often used:

~A ~{and} ~B _ = _ ~A &intersect. ~B

~A ~{or} ~B _ = _ ~A &union. ~B

~{not} ~A _ = _ ~A^c

Discrete Probability Spaces

If &Omega. is countable ( or finite ) then ( &Omega., @A, P ) is said to be a #~{discrete probability space}.
In this case we can assign probabilities to the points of &Omega., and this characterizes the space completely.
I.e. _ put _ &Omega. = \{ &omega._{~i} \}_{~i &in. &naturals.}, _ _ ~p_{~i} = P( \{ &omega._{~i} \} )
then

P( ~A ) _ _ _ = _ sum{ _ ~{p_i},\{ ~i | &omega._{~i} &in. ~A \}, _ }

as the \{ &omega._{~i} \} are disjoint

Examples

Roll of one die. &Omega. = \{ 1, 2, 3, 4, 5, 6 \}
If die is fair then ~p_1 = ~p_2 = ~p_3 = ~p_4 = ~p_5 = ~p_6 = 1/6

Roll of two dice. &Omega. = \{ ( 1,1 ), ( 1,2 ), ... , ( 6,5 ), ( 6,6 ) \}
In this case we assume that the dice are distinguishable, so that, e.g. ( 2, 5 ) &neq. ( 5, 2 )