![Normal Distribution CDF](https://upload.wikimedia.org/wikipedia/commons/c/ca/Normal_Distribution_CDF.svg)
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 o ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the cumulative distribution function (CDF) of a real-valued
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, or just distribution function of
, evaluated at
, is the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
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