Tag Info

Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.

The cumulative distribution function of a random variable gives the probability of that variable taking on a value less than or equal to any given value. The cdf of a distribution is the integral of the pdf of that distribution: $$F(x)=Pr(X\le x)=\int_{-\infty}^{x}f(\xi)d\xi$$ (If the random variable is discrete, this simplifies to a sum.)

In applied statistics, cdfs are important in comparing distributions, playing a role in plots (e.g., pp-plots), and hypothesis tests (e.g., the Kolmogorov-Smirnov test).