Cumulative Distribution Function From Density
Why
The density of a random variable determines the cumulative distribution function.
Result
A probability density
function of a random variable,
if it exists,
characterizes the cumulative
distribution function.
Let $(X, \mathcal{A} , \mu )$ be a
probability space.
Let $f$ be a real-valued
random variable on $X$.
Let $\lambda $ denote
the cover length.
Let $g$ be a probability
density of $f$:
Then:
\[ \begin{aligned} \rvcdf{f}(t) &= \mu (\Set*{x \in X}{f(x) \in (-\infty, t])}) \\ &= \int_{(-\infty, t]} gd\lambda . \end{aligned} \]