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

• AKA: Cumulative Continuous Probability, Cumulative Distribution Function.
• Context:
  • It can range from being a Theoretical CDF to being an Empirical CDF.
  • …
• Example(s):
  • Marginal Cumulative Probability Function.
  • …
• Counter-Example(s):
  • Joint Cumulative Probability Function.
• See: Continuous Random Variable, CDF Function Estimation.

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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 $X$ , or just distribution function of $X$ , evaluated at $x$ , is the probability that $X$ will take a value less than or equal to $x$ . 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 $F:\mathbb{R}\to [0,1]$ satisfying $\underset{x\to -\mathrm{\infty }}{lim}F(x)=0$ and $\underset{x\to \mathrm{\infty }}{lim}F(x)=1$ .

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

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),

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