Random variable: Difference between revisions

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=== Functions of random variables ===

If we have a random variable ''X'' on '''R''',Ω and a [[measurable function]] ''f'':'''R'''->'''R''', then naturally ''Y''=''f''(''X'') will also be a random variable on '''R'''Ω, since the composition of measurable functions is measurable. The same procedure that allowed one to go from a probability space (Ω,P) to ('''R''',dF_''X'') can be used to obtain the probability distribution of ''Y''.

The cumulative distribution function of ''Y'' is

:F_''Y''(''y'') = Prob(''f''(''X'')≤y).

The function ''f'' is a measurable function from the probability space ('''R''',dF_X); in other words, ''Y''=''f''(''X'') is a random variable. The cumulative distribution function of ''Y'' is

:F_''Y''(''y'')=Prob(''f''(''X'')≤y).

==== Example ====

Let ''f''(''x'')=''x''². Then,

:F_''Y''(''y'') = Prob(''X''²≤y).

If ''y''<0, then Prob(''X''^²&le;''y'')=0, so

:F_''Y''(''y'') = 0 if ''y''<0.

If ''y''=''a''²&ge;0, then Prob(''X''²&le;''y'')=Prob(|''X''|&le;&radic;''ay'')=Prob(-&radic;''ay''&le;''X''&le;&radic;''ay''), so

:F_''Y''(''y'') = F_''X''(&radic;''ay'')-F_''X''(-&radic;''ay'') if ''y''&ge;0.

=== Moments ===

A random variable is often characterised by a small number of quantities, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of [[expected value]] of a random variable, denoted E[''X'']. Note that in general, E[''f''(''X'')] is '''not''' the same as ''f''(E[''X'']). Once the "average value" is known, one could then ask how far from thethis average value the values of ''X'' typically are, a question that is answered by the [[variance]] and [[standard deviation]] of a random variable.

Mathematically, this is known as the (generalised) [[problem of moments]]: for a given class of random variables ''X'', find a collection {''f_i''} of functions such that the expectation values E[''f_i''(''X'')] fully characterize the distribution of the random variable ''X''.