Let $I$ be a countable set, and impose the counting measure on $I$ ($\mu(A):=|A|$, the cardinality of $A$, for any subset $A\subset I$). A probability distribution function on $I$ is then a nonnegative function $f:I\longrightarrow\mathbb{R}$ satisfying the equation

One example is the Poisson distribution $P_{r}$ on $\mathbb{N}$ (for any real number $r$), which is given by

for any $i\in\mathbb{N}$.

Given any probability space $(\Omega,\mathfrak{B},\mu)$ and any random variable $X:\Omega\longrightarrow I$, we can form a distribution function on $I$ by taking $f(i):=\mu(\{X=i\})$. The resulting function is called the distribution of $X$ on $I$.

Suppose $(\Omega,\mathfrak{B},\mu)$ equals $(\mathbb{R},\mathfrak{B}_{\lambda},\lambda)$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f:\mathbb{R}\longrightarrow\mathbb{R}$ is simply a measurable, nonnegative almost everywhere function such that

The associated measure has Radon–Nikodym derivative with respect to $\lambda$ equal to $f$:

One defines the cumulative distribution function $F$ of $f$ by the formula

for all $x\in\mathbb{R}$. A well known example of a probability distribution function on $\mathbb{R}$ is the Gaussian distribution, or normal distribution