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Probability Density Function

The probability density function (PDF) P(x) of a continuous distribution is defined as the derivative of the (cumulative) distribution function D(x),

D^'(x)=[P(x)]_(-infty)^x
(1)
=P(x)-P(-infty)
(2)
=P(x),
(3)

so

D(x)=P(X<=x)
(4)
=int_(-infty)^xP(xi)dxi.
(5)

A probability function satisfies

P(x in B)=int_BP(x)dx
(6)

and is constrained by the normalization condition,

P(-infty<x<infty)=int_(-infty)^inftyP(x)dx
(7)
=1.
(8)

Special cases are

P(a<=x<=b)=int_a^bP(x)dx
(9)
P(a<=x<=a+da)=int_a^(a+da)P(x)dx
(10)
approx P(a)da
(11)
P(x=a)=int_a^aP(x)dx
(12)
=0.
(13)

To find the probability function in a set of transformed variables, find the Jacobian. For example, If u=u(x), then

P_udu=P_xdx,
(14)

so

P_u=P_x|(partialx)/(partialu)|.
(15)

Similarly, if u=u(x,y) and v=v(x,y), then

P_(u,v)=P_(x,y)|(partial(x,y))/(partial(u,v))|.
(16)

Given n probability functions P_1(x), P_2(y), ..., P_n(z), the sum distribution X+Y+...+Z has probability function

P(t)=intintP_1(x)P_2(y)...P_n(z)delta((x+y+...+z)-t)dxdy...dz,
(17)

where delta(x) is a delta function. Similarly, the probability function for the distribution of XY...Z is given by

P(t)=intintP_1(x)P_2(y)...P_n(z)delta(xy...z-t)dxdy...dz.
(18)

The difference distribution X-Y has probability function

P(t)=intintP_1(x)P_2(y)delta((x-y)-t)dxdy,
(19)

and the ratio distribution X/Y has probability function

P(t)=intintP_1(x)P_2(y)delta((x/y)-t)dxdy,
(20)

Given the moments of a distribution (mu, sigma, and the gamma statistics gamma_r), the asymptotic probability function is given by

P(x)=Z(x)-[1/6gamma_1Z^((3))(x)]+[1/(24)gamma_2Z^((4))(x)+1/(72)gamma_1^2Z^((6))(x)]-[1/(120)gamma_3Z^((5))(x)+1/(144)gamma_1gamma_2Z^((7))(x)+1/(1296)gamma_1^3Z^((9))(x)]+[1/(720)gamma_4Z^((6))(x)+(1/(1152)gamma_2^2+1/(720)gamma_1gamma_3)Z^((8))(x)+1/(1728)gamma_1^2gamma_2Z^((10))(x)+1/(31104)gamma_1^4Z^((12))(x)]+...,
(21)

where

Z(x)=1/(sigmasqrt(2pi))e^(-(x-mu)^2/(2sigma^2))
(22)

is the normal distribution, and

gamma_r=(kappa_r)/(sigma^(r+2))
(23)

for r>=1 (with kappa_r cumulants and sigma the standard deviation; Abramowitz and Stegun 1972, p. 935).

See also

Continuous Distribution, Cornish-Fisher Asymptotic Expansion, Discrete Distribution, Distribution Function, Joint Distribution Function, Ratio 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 Resource. https://mathworld.wolfram.com/ProbabilityDensityFunction.html

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