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# Probability Density Function

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).

Continuous Distribution, Cornish-Fisher Asymptotic Expansion, Difference Distribution, Discrete Distribution, Distribution Function, Joint Distribution Function, Product Distribution, Ratio Distribution, Sum Distribution

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## References

Abramowitz, M. and Stegun, I. A. (Eds.). "Probability Functions." Ch. 26 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 925-964, 1972.Evans, M.; Hastings, N.; and Peacock, B. "Probability Density Function and Probability Function." §2.4 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 9-11, 2000.McLaughlin, M. "Common Probability Distributions." http://www.geocities.com/~mikemclaughlin/math_stat/Dists/Compendium.html.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 94, 1984.

## Referenced on Wolfram|Alpha

Probability Density Function

## Cite this as:

Weisstein, Eric W. "Probability Density Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProbabilityDensityFunction.html