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*functions of random variables are random variables. Example of cdf of X^2 |
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Mathematically, a random variable is defined as a [[measurable function]] from a [[probability space]] to some [[measurable space]]. This measurable space is the space of possible values of the variable, and it is usually taken to be the [[real number|real numbers]] with the [[Borel algebra|Borel σ-algebra]], and we will always assume this in this encyclopedia, unless otherwise specified.
=== Distribution functions ===
If a random variable ''X'':Ω->'''R''' defined on the probability space (Ω, ''P'') is given, we can ask questions like "How likely is it that the value of ''X'' is bigger than 2?". This is the same as the probability of the event {''s'' in Ω : ''X''(''s'') > 2} which is often written as ''P''(''X'' > 2) for short.
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and sometimes also using a [[probability density function]]. In [[measure theory|measure-theoretic]] terms, we use the random variable ''X'' to "push-forward" the measure ''P'' on Ω to a measure d''F'' on '''R'''.
The underlying probability space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just puts a measure on '''R''' that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.
=== Functions of random variables ===
If we have a random variable ''X'' on '''R''', and a measurable function ''f'':'''R'''->'''R''', then naturally ''Y''=''f''(''X'') will also be a random variable on '''R'''. The same procedure that allowed one to go from a probability space (Ω,P) to ('''R''',dF<sub>''X''</sub>) can be used to obtain the probability distribution of ''Y''.
The function ''f'' is a measurable function from the probability space ('''R''',dF<sub>X</sub>); in other words, ''Y''=''f''(''X'') is a random variable. The cumulative distribution function of ''Y'' is
:F<sub>''Y''</sub>(''y'')=Prob(''f''(''X'')≤y).
Example: Let ''f''(''x'')=''x''<sup>2</sup>. Then,
:F<sub>''Y''</sub>(''y'')=Prob(''X''<sup>2</sup>≤y).
If ''y''<0, then Prob(''X''^<sup>2</sup>le;y)=0, so
:F<sub>''Y''</sub>(''y'')=0 if ''y''<0.
If ''y''=''a''<sup>2</sup>≥0, then Prob(''X''<sup>2</sup>≤y)=Prob(|''X''|≤''a'')=Prob(-''a''≤''X''≤''a''), so
:F<sub>''Y''</sub>(''y'')=F<sub>''X''</sub>(''a'')-F<sub>''X''</sub>(-''a'') if ''y''≥0.
=== Moments ===
A random variable can often be 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. For instance, what is the average of the results you get when you roll a die represtedly, or measure human heights? This is captured by the mathematical concept of [[expected value]] of a random variable, denoted E[''X'']. Once the "average value" is known, one could then ask "how far are the values of ''X'' typically away from the average", a question that is answered by the [[variance]] and [[standard deviation]] of a random variable.
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