The probability density function (PDF) of a continuous distribution is defined as the derivative of the (cumulative) distribution function ,
(1)
 
(2)
 
(3)

so
(4)
 
(5)

A probability function satisfies
(6)

and is constrained by the normalization condition,
(7)
 
(8)

Special cases are
(9)
 
(10)
 
(11)
 
(12)
 
(13)

To find the probability function in a set of transformed variables, find the Jacobian. For example, If , then
(14)

so
(15)

Similarly, if and , then
(16)

Given probability functions , , ..., , the sum distribution has probability function
(17)

where is a delta function. Similarly, the probability function for the distribution of is given by
(18)

The difference distribution has probability function
(19)

and the ratio distribution has probability function
(20)

Given the moments of a distribution (, , and the gamma statistics ), the asymptotic probability function is given by
(21)

where
(22)

is the normal distribution, and
(23)

for (with cumulants and the standard deviation; Abramowitz and Stegun 1972, p. 935).