Probability Density Function

The probability density function (PDF) of a real random variable with an absolutely continuous cumulative distribution function is a nonnegative measurable function satisfying

(1)

almost everywhere, so

			(2)
			(3)

For every Borel set , a probability density function satisfies

(4)

and is constrained by the normalization condition,

			(5)
			(6)

For , special cases are

			(7)
			(8)
			(9)
			(10)
			(11)

The small-interval expansion uses little-O notation and assumes that is continuous at and .

To transform a probability density function, use the Jacobian of the inverse transformation. If is a continuously differentiable one-to-one map on an interval containing almost surely, with nonzero derivative, then

(12)

so

(13)

Similarly, if is a continuously differentiable diffeomorphism between open subsets of containing the random vectors almost surely, then the joint probability density functions satisfy

(14)

Given independent random variables with probability density functions , , , , the sum distribution has probability density function

(15)

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

(16)

The difference distribution has probability function

(17)

and the ratio distribution has probability function

(18)

For independent random variables with probability density functions and finite absolute moments through order 6, write , and . The density of the standardized sum has the following formal asymptotic series:

(19)

where

(20)

is the density of the standard normal distribution, and

(21)

for , where is the th cumulant of (Abramowitz and Stegun 1972, p. 935, formula 26.2.47). The derivatives are with respect to . These formal coefficients alone do not assert a density remainder estimate for arbitrary independent summands.

Probability density functions are computed in the Wolfram Language using PDF.