In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the cumulative distribution function (CDF) of a real-valued
random variable , or just distribution function of
, evaluated at
, is the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
that
will take a value less than or equal to
.
Every probability distribution
supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous''
''monotonic increasing'' cumulative distribution function