# Cumulative Density Function (CDF)

A Cumulative Density Function (CDF) is a non-decreasing right-continuous unit function that returns the probability that a real-valued random variable X (with a given probability distribution) will be found at a value less than or equal to x

## References

### 2022

• (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Cumulative_distribution_function Retrieved:2022-8-15.
• In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable $\displaystyle{ X }$ , or just distribution function of $\displaystyle{ X }$ , evaluated at $\displaystyle{ x }$ , is the probability that $\displaystyle{ X }$ will take a value less than or equal to $\displaystyle{ x }$ . 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 $\displaystyle{ F : \mathbb R \rightarrow [0,1] }$ satisfying $\displaystyle{ \lim_{x\rightarrow-\infty}F(x)=0 }$ and $\displaystyle{ \lim_{x\rightarrow\infty}F(x)=1 }$ .

In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to $\displaystyle{ x }$ . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.