Cumulative Distribution Function as Integral of Probability Density Function

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an absolutely continuous random variable.

Let $f_X$ be a probability density function for $X$.

Let $F_X$ be the cumulative distribution function for $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

Then:

$\ds \map {F_X} x = \int_{-\infty}^x f_X \rd \lambda$

for each $x \in \R$, where $\ds \int_{-\infty}^x f_X \rd \lambda$ denotes the Lebesgue integral of $f$ over $\hointl {-\infty} x$.

Proof

Let $P_X$ be the probability distribution of $X$.

Since $f_X$ is a probability density function for $X$, $f_X$ is a Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.

Then, we have:

\(\ds \map {F_X} x\)

\(=\)

\(\ds \map \Pr {X \le x}\)

Definition of Cumulative Distribution Function

\(\ds \)

\(=\)

\(\ds \map \Pr {X \in \hointl {-\infty} x}\)

\(\ds \)

\(=\)

\(\ds \map {P_X} {\hointl {-\infty} x}\)

Definition of Probability Distribution

\(\ds \)

\(=\)

\(\ds \int_{\hointl {-\infty} x} \map {f_X} t \rd \map \lambda t\)

Definition of Radon-Nikodym Derivative

\(\ds \)

\(=\)

\(\ds \int_{-\infty}^x f_X \rd \lambda\)

$\blacksquare$