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Random Variables

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Measurable Functions

Let ( E, @E ), ( F, @F ) be measurable spaces. A function ~X: E --> F is #~{measurable} ( w.r.t. @E , @F ) _ if ~X ^{-1}( A ) &in. @E, _ &forall. A &in. @F.
[Write ~X ^{-1}( @F ) &subset. @E ].

Conditions of Measurability

If ~X: E --> F is a function, and @C &subset. 2^F, &sigma.( @C ) = @F, _ then _ ~X is measurable ( w.r.t. @E , @F ) &iff. ~X ^{-1}( @C ) &subset. @E.

Proof
=> : _ obvious.
<= : _ Suppose ~X ^{-1}( C ) &in. @E, _ &forall. C &in. @C. _ Put @B = \{ Λ &in. @F | ~X ^{-1}( Λ ) &in. @E \}, _ then @C &in. @B, _ also @B is a &sigma.-algebra
[ since ~X ^{-1}( &union. Λ_{~i} ) = &union.( ~X ^{-1}( Λ_{~i} ) ); _ ~X ^{-1}( &intersect. Λ_{~i} ) = &intersect.( ~X ^{-1}( Λ_{~i} ) ); _ ~X ^{-1}( Λ^c ) = ( ~X ^{-1}( Λ ) )^c ],
so &sigma.( @C ) &subset. @B but &sigma.( @C ) = @F but @B &subset. @F so @B = @F.

Composite Measurable Functions

( E, @E ), ( F, @F ), ( G, @G ) measurable spaces, ~X : E --> F measurable ( w.r.t. @E , @F ) _ and _ ~Y : F --> G measurable ( w.r.t. @F , @G )
then _ ~Y o ~X : E --> G measurable ( w.r.t. @E , @G )

Random Variables

( &Omega., @A, P ) probability space, a #~{random variable} is a measurable function _ ~X: ( &Omega., @A ) --> ( &reals., @B ) _ where _ @B _ = _ @B( &reals. ) _ is the Borel &sigma.-algebra generated by the open sets of &reals..

&Omega. zDgrmRight{~X, _ } &reals.
@A zDgrmLeft{,~X ^{-1}} @B

A sufficient condition for ~X to be a random variable is the following:

A set function _ ~X: ( &Omega., @A ) --> ( &reals., @B( &reals. ) ) is measurable _ <=> _ ~X ^{-1}( ( {-&infty.}, ~x ] ) &in. @A, _ any ~x &in. &reals..
[with the convention that ( {-&infty.}, {-&infty.} ] = &empty. _ and _ ( {-&infty.}, &infty. ] = &reals.. ]

This is a special case of the general result for measurablility of functions.

Let @C _ be the collection of all half open intervals of the form ( {-&infty.}, ~c ]. We need to show that &sigma.( @C _ ) = @B( &reals. ) :

Let @K _ be the collection of all sub-sets K &subset. &reals. which are the countable union of open intervals, i.e. K = &union._{~i &in. &naturals.} ( ~{x_i}, ~{y_i} ], _ {-&infty.} &le. ~{x_i} &le. ~{y_i} &le. &infty., _ &forall. ~i. _ Putting ( ~x, &infty. ] = ( ~x, &infty. ) _ and _ ( ~x, ~y ] = &empty. when ~x = ~y. @K _ is an algebra and @C &subset. @K . So &sigma.( @C _ ) &subset. &sigma.( @K _ ).
Now _ ( ~x, ~y ] = ( {-&infty.}, ~y ] #{&bslash.} ( {-&infty.}, ~x ]

Law of Random Variable

~X: ( &Omega. , @A ) --> ( &reals. , @B ) random variable. The #~{distribution}, or #~{law} of ~X ( with respect to P ) is the function P ^~X : @B --> [0,1],

P ^~X ( A ) _ = _ P ( \{&omega. | ~X( &omega. ) &in. A\} ) _ = _ P ( ~X ^{-1}( A ) )

This is sometimes written in the form P( ~X &in. A ).

&Omega. zDgrmRight{~X, _ } &reals.
@A zDgrmLeft{ _ , ~X ^{-1}} @B
zDgrmDown{P,}   zDgrmVtLine{ _ ,P ^~X}
[ 0 , 1 ] zDgrmLeft{,}

Distribution Function

If ~X is a random variable _ ~X: ( &Omega., @A ) --> ( &reals., @B ), and P is a probability measure on ( &Omega., @A ), then we define the #~{(cumulative) distribution function} (#~{c.d.f.}) _ F_~X #: &reals. --> [0,1 ] _ by

F_~X ( ~c ) _ #:= _ P( ~X =< ~c ) _ #:= _ P ^~X ( ( {-&infty.}, ~c ] ) _ = _ P( ~X ^{-1}( ( {-&infty.}, ~c ] ) )

This definition makes sense due to the condition on the measurability of X above. In fact, by the same result we can say that the distribution function completely characterizes the random variable ( w.r.t. probability measure P ).

Properties of Distribution Functions

Any distribution function F has the following properties:

  1. F is non-decreasing: _ ~y > ~x ( &in. &reals. ) _ => _ F( ~y ) >= F( ~x )
  2. F is right continuous: _ ~x_{n} &downarrow. ~x _ => _ F( ~x_{n} ) &downarrow. F( ~x )
  3. lim_{~x -> {{-&infty.}}} F( ~x ) = 0; _ lim_{~x -> &infty.} F( ~x ) = 1;


Conversely, if any function F satisfies 1 - 3 , then we can find ( construct ) a unique probability measure Q on ( &Omega., @A ), so that F is the distribution function of ~X w.r.t. Q.

Function of a Random Variable

If ~X is a random variable, and &phi.: &reals. --> &reals. _ is a measurable function, _ then &phi.( ~X ) _ ( = &phi. &comp. ~X ) is also a random variable.

&Omega. zDgrmRight{~X, _ } &reals. zDgrmRight{&phi. , _ } &reals.
@A zDgrmLeft{ _ , ~X ^{-1}} @B zDgrmLeft{ _ , &phi.^{-1}} @B