Probability Function

The probability density function (also called the Probability Density Function) of a continuous distribution is defined as the derivative of the (cumulative) Distribution Function ,

$\begin{displaymath} D'(x) = [P(x)]_{-\infty}^x = P(x)-P(-\infty) = P(x), \end{displaymath}$

(1)

$\begin{displaymath} D(x) = P(X\leq x) \equiv \int^x_{-\infty} P(y)\,dy. \end{displaymath}$

(2)

A probability density function satisfies

$\begin{displaymath} P(x\in B) = \int_B P(x)\,dx \end{displaymath}$

(3)

and is constrained by the normalization condition,

$\begin{displaymath} P(-\infty < x < \infty) = \int_{-\infty}^\infty P(x)\,dx \equiv 1. \end{displaymath}$

(4)

Special cases are

$\displaystyle P(a\leq x\leq b)$	$\textstyle =$	$\displaystyle \int^b_a P(x)\,dx$	(5)
$\displaystyle P(a\leq x\leq a+da)$	$\textstyle =$	$\displaystyle \int^{a+da}_a P(x)\,dx \approx P(a)\,da$	(6)
$\displaystyle P(x = a)$	$\textstyle =$	$\displaystyle \int^a_a P(x)\,dx = 0.$	(7)

If and , then

$\begin{displaymath} P_{u,v}(u,v) = P_{x,y}(x,y)\left\vert{\partial (x,y)\over \partial (u,v)}\right\vert. \end{displaymath}$

(8)

Given the Moments of a distribution ( $\mu$ , $\sigma$ , and the Gamma Statistics $\gamma_r$ ), the asymptotic probability function is given by

$\quad -[{\textstyle{1\over 6}} \gamma_1 Z^{(3)}(x)]+[{\textstyle{1\over 24}} \gamma_2 Z^{(4)}(x)+{\textstyle{1\over 72}} {\gamma_1}^2 Z^{(6)}(x)]$

$\quad -[{\textstyle{1\over 120}} \gamma_3 Z^{(5)}(x)+{\textstyle{1\over 144}}\gamma_1\gamma_2 Z^{(7)}(x)+{\textstyle{1\over 1296}}{\gamma_1}^3 Z^{(9)}(x)]$

$\quad +[{\textstyle{1\over 720}} \gamma_4 Z^{(6)}(x)+({\textstyle{1\over 1152}}{\gamma_2}^2+{\textstyle{1\over 720}} \gamma_1\gamma_3)Z^{(8)}(x)$

$\quad +{\textstyle{1\over 1728}} {\gamma_1}^2\gamma_2 Z^{(10)}(x)+{\textstyle{1\over 31104}} {\gamma_1}^4 Z^{(12)}(x)]+\ldots,$ (9)

where

$\begin{displaymath} Z(x)={1\over\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/2\sigma^2} \end{displaymath}$

(10)

is the Normal Distribution, and

$\begin{displaymath} \gamma_r ={\kappa_r\over\sigma^{r+2}} \end{displaymath}$

(11)

for $r\geq 1$ (with $\kappa_r$ Cumulants and $\sigma$ the Standard Deviation; Abramowitz and Stegun 1972, p. 935).

References

Abramowitz, M. and Stegun, C. 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.


$\quad -[{\textstyle{1\over 6}} \gamma_1 Z^{(3)}(x)]+[{\textstyle{1\over 24}} \gamma_2 Z^{(4)}(x)+{\textstyle{1\over 72}} {\gamma_1}^2 Z^{(6)}(x)]$
$\quad -[{\textstyle{1\over 120}} \gamma_3 Z^{(5)}(x)+{\textstyle{1\over 144}}\gamma_1\gamma_2 Z^{(7)}(x)+{\textstyle{1\over 1296}}{\gamma_1}^3 Z^{(9)}(x)]$
$\quad +[{\textstyle{1\over 720}} \gamma_4 Z^{(6)}(x)+({\textstyle{1\over 1152}}{\gamma_2}^2+{\textstyle{1\over 720}} \gamma_1\gamma_3)Z^{(8)}(x)$
$\quad +{\textstyle{1\over 1728}} {\gamma_1}^2\gamma_2 Z^{(10)}(x)+{\textstyle{1\over 31104}} {\gamma_1}^4 Z^{(12)}(x)]+\ldots,$	(9)