probability density function

概率密度

Probability · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: PDF · pdf · 概率密度函数

Translation evidence: Literature usage

Meaning in context

A nonnegative measurable function $f$ with $P(X\in B)=\int_B f(x)\,dx$ for every Borel set $B$, when the law is absolutely continuous with respect to Lebesgue measure.

当分布对勒贝格测度绝对连续时，用在集合上的积分给出概率的非负可测函数。

Example

Uniform probability on $[0,1/2]$ has density $2$ on that interval.

Usage and distinctions

A density may exceed one and is determined only almost everywhere; a continuous CDF need not have a density.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on ProbabilitySection 2, Random variables, density and Cantor-function example · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程（北京大学）§1.2.3 常用分布，连续型随机变量与概率密度 · Checked 2026-10-14

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