Random variable: Difference between revisions

Revision as of 23:07, 9 May 2004

Revision as of 23:10, 9 May 2004

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:<math>P[X\le x]=P[Y\le x]\quad\hbox{for all}\quad x.</math>

To be equal in distribution, random variables need not be defined on the same probability space, but without loss of generality they can be made into independent random variables on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between probability distributions,

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

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

Line 62:

:<math>P[X\neq Y]=0</math>

For all practical purposes in probability theory, this notion of equivalence is as strong as actuall equality. It is associated to the following distance:

:<math>d_\infty(X,Y)={\rm esssup}_sup_\omega|X(\omega)-Y(\omega)|.</math>

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

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