Random variable: Difference between revisions

Revision as of 23:10, 9 May 2004

Revision as of 23:37, 9 May 2004

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There are saveral different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, equal in mean, or equal in distribution.

In incresingincreasing order of strenght, the precise definition of these notions of equivalence is:

==== Equality in distribution ====

Two random variables ''X'' and ''Y'' are ''equal in distribution'' if

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:<math>d(X,Y)=\sup_x|P[X\le x]-P[Y\le x]|,</math>

which is the basis of the [[Kolmogorov-Smirnov test]].

==== Equality in mean ====

Two random variables ''X'' and ''Y'' are ''equal in p-th mean'' if the ''p''th moment of |''X''-''Y''| is zero, that is,

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Equality in ''p''th mean implies equality in ''q''th mean for all ''q''<''p''. As in the previous case, there is a related distance between the random variables, namely

:<math>d_p(X,Y)=E[|X-Y|^p].</math>

==== Almost sure equality ====

Two random variables ''X'' and ''Y'' are ''equal almost surely'' if, and only if, the probability that they are different is zero:

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:<math>d_\infty(X,Y)=\sup_\omega|X(\omega)-Y(\omega)|</math>

where 'sup' in this case represents the [[essential supremum]] in the sense of [[measure theory]].

==== Equality ====

Finally, two random variables ''X'' and ''Y'' are ''equal'' if they are equal as functions on their probability space, that is,