# Cumulative Density Function (CDF)

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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*

**AKA:**Cumulative Continuous Probability, Cumulative Distribution Function.**Context:**- It can range from being a Theoretical CDF to being an Empirical CDF.
- …

**Example(s):****Counter-Example(s):****See:**Continuous Random Variable, CDF Function Estimation.

## 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 [math]\displaystyle{ X }[/math] , or just**distribution function**of [math]\displaystyle{ X }[/math] , evaluated at [math]\displaystyle{ x }[/math] , is the probability that [math]\displaystyle{ X }[/math] will take a value less than or equal to [math]\displaystyle{ x }[/math] . 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 [math]\displaystyle{ F : \mathbb R \rightarrow [0,1] }[/math] satisfying [math]\displaystyle{ \lim_{x\rightarrow-\infty}F(x)=0 }[/math] and [math]\displaystyle{ \lim_{x\rightarrow\infty}F(x)=1 }[/math] .In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to [math]\displaystyle{ x }[/math] . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

- In probability theory and statistics, the

### 2006

- (Dubnicka, 2006c) ⇒ Suzanne R. Dubnicka. (2006). “Random Variables - STAT 510: Handout 3." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
- TERMINOLOGY : The cumulative distribution function (cdf) of a random variable X, denoted by FX(x), is given by FX(x) = P(X x), for all x 2 R.
- ALTERNATE DEFINITION: A random variable is said to be continuous if its cdf FX(x) is a continuous function of x.
- TERMINOLOGY : Let X be a continuous random variable with cdf FX(x). The probability density function (pdf) for X, denoted by fX(x), is given by fX(x) = d/dx FX(x),