probability distribution

A probability distribution is a function which maps each possible output of a random variable to the probability of the event it represents.

Formally,
consider an experiment with sample space $\mathrm{\Omega }$, equipped with event space $\mathcal{F}$.
Let $P:\mathcal{F}\to [0,1]$ be the probability measure.
Let $X:\mathrm{\Omega }\to R$ be a random variable to the measurable space $(R,\mathcal{R})$.
Then the probability distribution of $X$ is a function given by the pushforward measure of $P$ onto $(R,\mathcal{R})$ induced by $X$, given formally by

for every $I\in \mathcal{R}$, such that $X^{-1}(I)$ is an element of $\mathcal{F}$.

Typically, the set of real numbers $\mathbb{R}$ is used for $R$.
When $X$ is a discrete random variable, the probability distribution is a probability mass function.
When the random variable is continuous, the probability distribution is a probability density function.

A distribution is typically specified using the following notation:

for discrete random variables, and

for continuous random variables.

"It is the translation layer between the languages of topology and statistics."