Probability Density Function
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Probability Functions." Ch. 26 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
15-3-2021
2296
Probability Density Function
The probability density function (PDF) 
of a continuous distribution is defined as the derivative of the (cumulative) distribution function 
,
so
A probability function satisfies
 |
(6)
|
and is constrained by the normalization condition,
Special cases are
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).
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.
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