🎲 Cumulative Distribution Function (CDF)

Cumulative Distribution Function (CDF)#

The probability that a variable is at most a given value.

Important

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What it is#

The cumulative distribution function (CDF) gives the probability that a random variable is at most \(x\):

\[F(x) = \Pr[X \le x].\]

Unlike the PMF or PDF, it works for both discrete and continuous variables.

Properties#

The CDF is non-decreasing, runs from 0 to 1, and is right-continuous. For a discrete variable it is a step function; for a continuous one it is smooth:

\[F(x) = \sum_{k \le x} p(k) \quad\text{(discrete)}, \qquad F(x) = \int_{-\infty}^{x} f(t)\,dt \quad\text{(continuous)}.\]

Why it’s useful#

It directly answers “at most” and interval questions, and it is the bridge between representations: quantiles are read off its inverse and the density is its derivative. For a fair die, \(F(2) = 1/3\).

Theme: Probability & Statistics Foundations · All terminology

Hint

Mind map — connected ideas

Probability Density · Probability Mass · Probability Distribution · Quantile Level · Quantile Regression · Normal Distribution

Hint

More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion · Probability