# Cumulative distribution function

## Probability that random variable X is less than or equal to x / From Wikipedia, the free encyclopedia

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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 $x$.^{[1]}

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Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) $F\colon \mathbb {R} \rightarrow [0,1]$ satisfying $\lim _{x\rightarrow -\infty }F(x)=0$ and $\lim _{x\rightarrow \infty }F(x)=1$.

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