Real random variable

Cumulative distribution function

A cumulative distribution function (CDF) may be defined for either a Discrete random variable or a Continuous random variable such that

This can be used to find the probability of any range

The cumulative distribution function has the following properties

1. which follows from its definition in the context of the Probability model.
2. is monotone increasing, i.e. , see below.
3. , see below.
4. , see below.
5. Continuity varies between the discrete and continuous case, see below.

Random variables sharing the same CDF form equivalence classes under . See Variable distribution equivalence.

A useful property of the CDF is that given any variable with CDF

where is the standard uniform distribution. This allows for Inverse transform sampling.

Discrete variable

For a Discrete random variable, the cumulative distribution function is defined using the probability mass function such that

For a discrete variable is right-continuous.

Continuous variable

For a Continuous random variable, the cumulative distribution function is defined using the probability density function such that

For a continuous variable is continuous.

Inverse

See Quantile function.

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