Cumulative Distribution Functions
Definition
The cumulative distribution function of a real-valued random variable is the function mapping a real number to the measure of the set of outcomes for which the random variable takes value less than or equal to the number. Notation (below) helps.
The range of the cumulative distribution function is the interval $[0, 1]$, since the measure of the base set is one and all measures are non-negative.
We often abbreviate the words “cumulative distribution function” by c.d.f. or cdf.
Notation
Let $(X, \mathcal{A} , \mu )$ be a probability
space.
Let $f: X \to \R $ be a measurable function
(a real-valued random variable).
We denote the cumulative distribution function
of $f$ by $\rvcdf{f}: \R \to [0, 1]$.
We defined it by
\[
\rvcdf{f}(t) = \mu (\Set*{x \in X}{f(x) \leq t}).
\]
Properties
- is non-decreasing,
- is right-continuous,
- tends to one as its argument tends toward infinity, and
- tends to zero as its argument tends towards negative infinity.