The probability density function (PDF) of a continuous distribution is defined as the derivative of the (cumulative) distribution function ,
|
(1) | |||
|
(2) | |||
|
(3) |
so
A probability function satisfies
|
(6) |
and is constrained by the normalization condition,
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).