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 .
Read more about this topic: Random Variable
Famous quotes containing the words functions of, functions, random and/or variables:
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny fingerto let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (17701850)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny fingerto let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (17701850)
“poor Felix Randal;
How far from then forethought of, all thy more boisterous years,
When thou at the random grim forge, powerful amidst peers,
Didst fettle for the great gray drayhorse his bright and battering
sandal!”
—Gerard Manley Hopkins (18441889)
“Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
—Paul Valéry (18711945)