cumulative distribution function

分布函数

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

Other names / variants: CDF · cdf

Translation evidence: Literature usage

Meaning in context

For a real random variable $X$, $F_X(x)=P(X\le x)$; it is nondecreasing, right-continuous, and has limits $0$ and $1$ at the two infinities.

实随机变量不超过x的概率所构成的函数；它单调不减、右连续，两端极限分别为0和1。

Example

For $X=0$ almost surely, $F_X(x)=0$ for $x<0$ and $1$ for $x\ge0$.

Usage and distinctions

The convention uses a non-strict inequality; a jump is the mass at that point.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on ProbabilitySection 2, Random variables, distribution function and its properties · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程（北京大学）§1.2.1 定义1.6；§1.2.2 分布函数的性质 · Checked 2026-10-14

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