 # Cumulative distribution function

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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 $x$ .
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) $F:\mathbb {R} \rightarrow [0,1]$ 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 minus infinity to $x$ . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.