GV

GeezerVison

The World Wide Web, restored to 800 pixels of glory.

Home|About|Gaming cabinet|Archives|How it works|Starter sites

On this site

  • Algebra
  • Applied Mathematics
  • Calculus and Analysis
  • Discrete Mathematics
  • Foundations of Mathematics
  • Geometry
  • History and Terminology
  • Number Theory
  • Probability and Statistics
  • Recreational Mathematics
  • Topology
  • Alphabetical Index
  • Random Entry
  • New in MathWorld

This view

  • Original page
  • New address

GeezerVison is showing mathworld.wolfram.com — Probability Density Function -- from Wolfram MathWorld

Search

Probability Density Function

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

D^'(x)=P(x),
(1)

almost everywhere, so

D(x)=P(X<=x)
(2)
=int_(-infty)^xP(xi)dxi.
(3)

For every Borel set B, a probability density function satisfies

P(X in B)=int_BP(x)dx
(4)

and is constrained by the normalization condition,

P(-infty<X<infty)=int_(-infty)^inftyP(x)dx
(5)
=1.
(6)

For a<=b, special cases are

P(a<=X<=b)=int_a^bP(x)dx
(7)
P(a<=X<=a+da)=int_a^(a+da)P(x)dx
(8)
=P(a)da+o(da)
(9)
P(X=a)=int_a^aP(x)dx
(10)
=0.
(11)

The small-interval expansion uses little-O notation and assumes that P is continuous at a and da->0^+.

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

P_u(u)|du|=P_x(x)|dx|,
(12)

so

P_u(u)=P_x(x(u))|(dx)/(du)|.
(13)

Similarly, if (u,v)=(u(x,y),v(x,y)) is a continuously differentiable diffeomorphism between open subsets of R^2 containing the random vectors almost surely, then the joint probability density functions satisfy

P_(u,v)(u,v)=P_(x,y)(x(u,v),y(u,v))|(partial(x,y))/(partial(u,v))|.
(14)

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

P(t)=∬P_1(x)P_2(y)...P_n(z)delta((x+y+...+z)-t)dxdy...dz,
(15)

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

P(t)=∬P_1(x)P_2(y)...P_n(z)delta(xy...z-t)dxdy...dz.
(16)

The difference distribution X-Y has probability function

P(t)=∬P_1(x)P_2(y)delta((x-y)-t)dxdy,
(17)

and the ratio distribution X/Y has probability function

P(t)=∬P_1(x)P_2(y)delta((x/y)-t)dxdy,
(18)

For independent random variables Y_1,...,Y_n with probability density functions and finite absolute moments through order 6, write mu_i=E(Y_i), sigma_i^2=Var(Y_i) and s^2=sum_(i=1)^(n)sigma_i^2>0. The density P_n(x) of the standardized sum sum_(i=1)^(n)(Y_i-mu_i)/s has the following formal asymptotic series:

P_n(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)]+...,
(19)

where

Z(x)=1/(sqrt(2pi))e^(-x^2/2)
(20)

is the density of the standard normal distribution, and

gamma_r=(sum_(i=1)^(n)kappa_(r+2,i))/(s^(r+2))
(21)

for r=1,...,4, where kappa_(r+2,i) is the (r+2)th cumulant of Y_i (Abramowitz and Stegun 1972, p. 935, formula 26.2.47). The derivatives Z^((j)) are with respect to x. 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.

See also

Continuous Distribution, Cornish-Fisher Asymptotic Expansion, Discrete Distribution, Distribution Function, Joint Distribution Function, Probability Mass Function, Ratio Distribution

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • birthday problem
  • Bayes' theorem
  • probability

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." https://www.oocities.org/~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

Subject classifications

You are visitor 01402503 — GeezerVison does not store the pages you look up.
Best viewed with Netscape Navigator 4.7 or Internet Explorer 6 at 800×600. No cookies required.