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## 10.7 Cumulative Distribution Functions

For most of the univariate probability functions, there is a corresponding cumulative distribution function, log cumulative distribution function, and log complementary cumulative distribution function.

For a univariate random variable $$Y$$ with probability function $$p_Y(y \, | \, \theta)$$, the cumulative distribution function (CDF) $$F_Y$$ is defined by $F_Y(y) \ = \ \text{Pr}[Y < y] \ = \ \int_{-\infty}^y p(y \, | \, \theta) \ \text{d}y.$ The complementary cumulative distribution function (CCDF) is defined as $\text{Pr}[Y \geq y] \ = \ 1 - F_Y(y).$ The reason to use CCDFs instead of CDFs in floating-point arithmetic is that it is possible to represent numbers very close to 0 (the closest you can get is roughly $$10^{-300}$$), but not numbers very close to 1 (the closest you can get is roughly $$1 - 10^{-15}$$).

In Stan, there is a cumulative distribution function for each probability function. For instance, normal_cdf(y, mu, sigma) is defined by $\int_{-\infty}^y \text{Normal}(y \, | \, \mu, \sigma) \ \text{d}y.$ There are also log forms of the CDF and CCDF for most univariate distributions. For example, normal_lcdf(y | mu, sigma) is defined by $\log \left( \int_{-\infty}^y \text{Normal}(y \, | \, \mu, \sigma) \ \text{d}y \right)$ and normal_lccdf(y | mu, sigma) is defined by $\log \left( 1 - \int_{-\infty}^y \text{Normal}(y \, | \, \mu, \sigma) \ \text{d}y \right).$