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Also known as: CDF
A function giving the probability that a random variable is less than or equal to a given value.
DESCRIPTION
The cumulative distribution function F(x) for a random variable X is defined as P(X ≤ x). CDFs fully characterize a distribution and enable probability statements, quantiles, and goodness-of-fit tests. Empirical CDFs estimate distributions from data. CDFs relate to PDFs (continuous) and PMFs (discrete), and to survival and quantile functions.
ALTERNATIVE DEFINITIONS
NIST Engineering Statistics Handbook
The cumulative distribution function (CDF) of a random variable X is the probability that X will take a value less than or equal to x: F(x) = P(X ≤ x).
Provides formal definition and notation.
Wolfram MathWorld
For a random variable X, the cumulative distribution function F(x) gives the probability that X is less than or equal to x.
Concise definition with mathematical focus.
Wikipedia
In probability theory and statistics, the cumulative distribution function of a real-valued random variable X, or just distribution function of X, is the probability that X will take a value less than or equal to x.
Adds terminology and context in probability and statistics.
RELATED TERMS
Quantile Function Survival Function Empirical CDF Distribution Function Random Variable Continuous Distribution
COMPETING TERMS
Probability Density Function Probability Mass Function
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist Data Analyst AI Engineer
EXTERNAL SOURCES
- Wikipedia - Cumulative distribution function
- NIST - Formal definition and examples
- StatLect - Detailed properties and proofs
TOP QUESTIONS
Why is the Cumulative Distribution Function important?
CDFs fully describe distributions and enable probability calculations, quantiles, and statistical testing.
Category: Relevancy
How does the CDF relate to the PDF?
For continuous variables, the PDF is the derivative of the CDF; integrating the PDF yields the CDF.
Category: Misunderstanding
What is an empirical CDF?
It estimates the CDF from observed data, stepwise increasing at each sample point.
Category: Definition
How do you compute quantiles from a CDF?
Use the inverse CDF (quantile function) to map probabilities to values.
Category: HowTo
What is the survival function?
It is 1 − F(x), the probability the variable exceeds x, used in reliability and survival analysis.
Category: Overview
What is a common mistake interpreting CDFs?
Confusing F(x) with density; F(x) is cumulative probability, not likelihood at x.
Category: Mistakes
Can CDFs be discontinuous?
Yes, for discrete distributions CDFs are step functions.
Category: Opinion
How are CDFs used in hypothesis testing?
They underpin tests like Kolmogorov–Smirnov and goodness-of-fit comparisons.
Category: Example
Where to learn about CDFs?
Consult NIST handbook, StatLect, and standard probability textbooks.
Category: Overview