Random variable: Difference between revisions

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=== Equivalence of random variables ===

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

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

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

To be equal indistribution, random variables need not be defined on the same probability space, but wothout loss of generality they can be made into independent random variables on the same probability space.

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

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

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

To be equal indistributionin distribution, random variables need not be defined on the same probability space, but wothoutwithout 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 y]|,</math>

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

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

:<math>E[|X-Y|^p]=0.</math>

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>

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

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

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

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

(see [[essential supremum]])

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