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The Probability Functions module provides cumulative distribution, density/mass, and quantile functions for a wide range of probability distributions.

Unless otherwise documented, all of these functions are wrappers around functionality provided by the boost C++ library [1, “Statistical Distributions and Functions”].

For convenience, all cumulative distribution and density/mass functions (CDFs and PDF/PMFs in short) are defined over the range of all floating-point numbers including infinity. Inputs that are NULL or NaN (not a number) will always produce a NULL or NaN result, respectively. Inputs that are plus or minus infinity will return the respective limits.

A quantile function for a probability distrution with CDF $ F $ takes a probability argument $p \in [0,1]$ and returns the value $ x $ so that $ F(x) = p $ , provided such an $ x $ exists and it is unique. If it does not, the result will be $\sup \{ x \in D \mid F(x) \leq p \}$ (interpreted as 0 if the supremum is over an empty set) if $ p < 0.5 $ , and $\inf \{ x \in D \mid F(x) \geq p \}$ if $p \geq 0.5$ . Here $ D $ denotes the domain of the distribution, which is the set of reals $\mathbb R$ for continuous and the set of nonnegative integers $\mathbb N_0$ for discrete distributions.

Intuitively, the formulas in the previous paragraph deal with the following special cases. The 0-quantile will always be the “left end” of the support, and the 1-quantile will be the “right end” of the support of the distribution. For discrete distributions, most values of $p \in [0,1]$ do not admit an $ x $ with $ F(x) = p $ . Instead, there is an $x \in \mathbb N_0$ so that $ F(x) < p < F(x + 1) $ . The above formulas mean that the value returned as $ p $ -quantile is $ x $ if $ p < 0.5 $ , and it is $ x + 1 $ if $p \geq 0.5$ . (As a special case, in order to ensure that quantiles are always within the support, the $ p $ -quantile will be 0 if $ p < F(0) $ ).

The rationale for choosing this behavior is that $p$ -quantiles for $ p < 0.5 $ are typically requested when interested in the value $ x $ such that with confidence level at least $ 1 - p $ a random variable will be $ > x $ (or equivalently, with probability at most $ p $ , it will be $\leq x$ ). Likewise, $p$ -quantiles for $p \geq 0.5$ are typically requested when interested in the value $ x $ such that with confidence level at least $ p $ a random variable will be $\leq x$ . See also [1, “Understanding Quantiles of Discrete Distributions”].

Function Syntax

Cumulative distribution functions:

distribution_cdf(random variate[, parameter1 [, parameter2 [, parameter3] ] ])

Probability density/mass functions:

distribution_{pdf|pmf}(random variate[, parameter1 [, parameter2 [, parameter3] ] ])

Quantile functions:

distribution_quantile(probability[, parameter1 [, parameter2 [, parameter3] ] ])

For concrete function signatures, see prob.sql_in.

Examples

SELECT madlib.normal_cdf(0);

Result:

 normal_cdf
 -----------
        0.5

SELECT madlib.normal_quantile(0.5, 0, 1);

Result:

 normal_quantile
 ----------------
               0
(1 row)

Literature

[1] John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg: Boost Math Toolkit, Version 1.49, available at: http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/index.html

Related Topics: File prob.sql_in documenting the SQL functions.