# Generating Random Numbers According to a Continuous Probability Distribution with PostgreSQL

When working with statistics, generating pseudo-random samples that are distributed according to a given probability distribution is a frequent requirement. For me, being able to generate these numbers directly with SQL would be very convenient (and might also improve performance). In this post, I’m taking a look at some of the most important continuous probability distributions…

## Continuous uniform distribution

Uniform distributions are easy, because we can use PostgreSQL’s built-in RANDOM() function. RANDOM() returns a uniformly distributed DOUBLE PRECISION value in the range 0.0 <= x < 1.0. Adjusting that range is simply a question of adding and/or multiplying with appropriate values:

 1 2 -- random double precision value in the range 1.0 <= x < 20.0 SELECT 1 + 19 * RANDOM(); 

That being said, it is easy to encapsulate that behavior in a SQL function for later reuse:

 1 2 3 4 5 6 7 8 9 10 11 12 13 -- generates count uniformly distributed random -- numbers in the range min <= x < max CREATE OR REPLACE FUNCTION random_uniform( count INTEGER DEFAULT 1, min DOUBLE PRECISION DEFAULT 0.0, max DOUBLE PRECISION DEFAULT 1.0 ) RETURNS SETOF DOUBLE PRECISION RETURNS NULL ON NULL INPUT AS $$BEGIN RETURN QUERY SELECT min + (max - min) * RANDOM() FROM GENERATE_SERIES(1, count); END;$$ LANGUAGE plpgsql; 

The function’s relative frequency-histogram (based on 106 values):

Relative frequencies have been obtained as follows: SELECT c, COUNT(r) / 1000000.0 FROM GENERATE_SERIES(0,49) AS c LEFT OUTER JOIN random_uniform(1000000, 0, 50) AS r ON c = FLOOR(r) GROUP BY c ORDER BY c;

## Exponential distribution

RANDOM() is the only way of generating random numbers supported by PostgreSQL. Therefore, the exponential distribution is more challenging, because we must somehow derive these numbers from uniformly distributed numbers.

Since the cumulative distribution function of the exponential distribution is invertible, we can use a technique known as inverse transform sampling. All of this boils down to a simple mathematical expression, that we can encapsulate in another SQL function:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -- generates count exponentially distributed random -- numbers with given expected value CREATE OR REPLACE FUNCTION random_exp( count INTEGER DEFAULT 1, mean DOUBLE PRECISION DEFAULT 1.0 ) RETURNS SETOF DOUBLE PRECISION RETURNS NULL ON NULL INPUT AS $$DECLARE u DOUBLE PRECISION; BEGIN WHILE count > 0 LOOP u = RANDOM(); -- range: 0.0 <= u < 1.0 IF u != 0.0 THEN RETURN NEXT -LN(u) * mean; count = count - 1; END IF; END LOOP; END;$$ LANGUAGE plpgsql; 

The function’s relative frequency-histogram (based on 106 values):

Relative frequencies have been obtained as follows: SELECT c, COUNT(r) / 1000000.0 FROM GENERATE_SERIES(0,49) AS c LEFT OUTER JOIN random_exp(1000000, 5) AS r ON c = FLOOR(r) GROUP BY c ORDER BY c;

## Normal distribution

As far as continuous distributions are concerned, the normal distribution is probably the most important one.

To generate normally distributed numbers, we must again derive them from uniformly distributed numbers. Fortunately, mathematicians have discovered many ways of doing exactly that: Central limit theorem, Box-Muller transform, Marsaglia polar method, Ziggurat algorithm, …

For now, I’ve decided to implement the Marsaglia polar method, because it’s reasonably simple to implement in SQL:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 -- generates count normally distributed random numbers -- with given mean and given standard deviation CREATE OR REPLACE FUNCTION random_normal( count INTEGER DEFAULT 1, mean DOUBLE PRECISION DEFAULT 0.0, stddev DOUBLE PRECISION DEFAULT 1.0 ) RETURNS SETOF DOUBLE PRECISION RETURNS NULL ON NULL INPUT AS $$DECLARE u DOUBLE PRECISION; v DOUBLE PRECISION; s DOUBLE PRECISION; BEGIN WHILE count > 0 LOOP u = RANDOM() * 2 - 1; -- range: -1.0 <= u < 1.0 v = RANDOM() * 2 - 1; -- range: -1.0 <= v < 1.0 s = u^2 + v^2; IF s != 0.0 AND s < 1.0 THEN s = SQRT(-2 * LN(s) / s); RETURN NEXT mean + stddev * s * u; count = count - 1; IF count > 0 THEN RETURN NEXT mean + stddev * s * v; count = count - 1; END IF; END IF; END LOOP; END;$$ LANGUAGE plpgsql; 

The function’s relative frequency-histogram (based on 106 values), clearly suggests that the generated numbers are indeed normally distributed (with expected value of 25 and standard deviation of 7):

Relative frequencies have been obtained as follows: SELECT c, COUNT(r) / 1000000.0 FROM GENERATE_SERIES(0,49) AS c LEFT OUTER JOIN random_normal(1000000, 25, 7) AS r ON c = FLOOR(r) GROUP BY c ORDER BY c;

Notes
Software: PostgreSQL v9.3.1