Functions of Random Variables
A new random variable Y can be defined by applying a real Borel measurable function to the outcomes of a real-valued random variable X. The cumulative distribution function of is
If function g is invertible, i.e. g−1 exists, and increasing, then the previous relation can be extended to obtain
and, again with the same hypotheses of invertibility of g, assuming also differentiability, we can find the relation between the probability density functions by differentiating both sides with respect to y, in order to obtain
- .
If there is no invertibility of g but each y admits at most a countable number of roots (i.e. a finite, or countably infinite, number of xi such that y = g(xi)) then the previous relation between the probability density functions can be generalized with
where xi = gi-1(y). The formulas for densities do not demand g to be increasing.
In the measure-theoretic, axiomatic approach to probability, if we have a random variable on and a Borel measurable function, then will also be a random variable on, since the composition of measurable functions is also measurable. (However, this is not true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .
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