Weibull distribution

Probability distribution
name =Weibull (2-Parameter)
type =density
pdf_

cdf_

parameters =$lambda>0,$ scale (real) $k>0,$ shape (real)
support =$x\; in\; [0;\; +infty),$
pdf = & xgeq0\0 & x<0end{cases}
cdf =$1-\; e^\{-(x/lambda)^k\}$
mean =$lambda\; Gammaleft(1+frac\{1\}\{k\}\; ight),$
median =$lambdaln(2)^\{1/k\},$
mode =$lambda\; left(frac\{k-1\}\{k\}\; ight)^\{frac\{1\}\{k,$ if $k>1$
arg mode =$lambdafrac\{k-1\}\{k\}^\{frac\{1\}\{k,$ if $k>1$
variance =$lambda^2Gammaleft(1+frac\{2\}\{k\}\; ight)\; -\; mu^2,$
skewness =$frac\{Gamma(1+frac\{3\}\{k\})lambda^3-3musigma^2-mu^3\}\{sigma^3\}$
kurtosis =(see text)
entropy =$gammaleft(1!-!frac\{1\}\{k\}\; ight)+lnleft(frac\{lambda\}\{k\}\; ight)+1$
mgf = see Weibull fading
char = see [Coastal Engineering,(2007),54(8),pp630- 638;doi:10.1016/j.coastaleng.2007.05.001]

In probability theory and statistics, the Weibull distribution [Weibull, W. (1951) "A statistical distribution function of wide applicability" "J. Appl. Mech.-Trans. ASME" 18(3), 293-297] (named after Waloddi Weibull) is a continuous probability distribution. It is often called the Rosin–Rammler distribution when used to describe the size distribution of particles. The distribution was introduced by P. Rosin and E. Rammler in 1933. [http://www.zarm.uni-bremen.de/gamm98/num_abs/a912.html] The probability density function of a Weibull random variable x is [Papoulis, Pillai, "Probability, Random Varibles, and Stochastic Processes, 4th Edition] :

:

where $k\; >0$ is the "shape parameter" and $lambda\; >0$ is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential.

The Weibull distribution is often used in the field of life data analysis due to its flexibility&mdash;it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then "k" < 1. If the failure rate is constant over time, then "k" = 1. If the failure rate increases over time, then "k" > 1.

An understanding of the failure rate may provide insight as to what is causing the failures:

* A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.

* A constant failure rate suggests that items are failing from random events.

* An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.

When "k" = 1, the Weibull distribution reduces to the exponential distribution.When "k" = 3.4, the Weibull distribution appears similar to the normal distribution.

Properties

There is an abrupt change in the value of the density function at 0 when $k$ takes on values around 1. It is because, [http://www.weibull.com/LifeDataWeb/characteristics_of_the_weibull_distribution.htm]
* for any as $x\; ightarrow\; 0$
* for $k=1,\; f(0)=frac\{1\}\{lambda\}$, and
* for any $k\; >\; 1,\; f(x)\; ightarrow\; 0$ as $x\; ightarrow\; 0$

The "n"th raw moment is given by:

:$m\_n\; =\; lambda^n\; Gamma(1+frac\{n\}\{k\}),$

where $Gamma$ is the Gamma function. The mean and variance of a Weibull random variable can be expressed as:

:$mathrm\{E\}(X)\; =\; lambda\; Gamma(1+frac\{1\}\{k\}),$

and

:$extrm\{var\}(X)\; =\; lambda^2\; [Gamma(1+frac\{2\}\{k\})\; -\; Gamma^2(1+frac\{1\}\{k\})]\; ,.$

The skewness is given by:

:$gamma\_1=frac\{Gammaleft(1+frac\{3\}\{k\}\; ight)lambda^3-3musigma^2-mu^3\}\{sigma^3\}.$

The excess kurtosis is given by:

:$gamma\_2=frac\{-6Gamma\_1^4+12Gamma\_1^2Gamma\_2-3Gamma\_2^2-4Gamma\_1Gamma\_3+Gamma\_4\}\{\; [Gamma\_2-Gamma\_1^2]\; ^2\}$

where $Gamma\_i=Gamma(1+i/k)$. The kurtosis excess may also be written as :

:$gamma\_\{2\}=frac\{lambda^4Gamma(1+frac\{4\}\{k\})-4gamma\_\{1\}sigma^3mu-6mu^2sigma^2-mu^4\}\{sigma^4\}$

A generalized, 3-parameter Weibull distribution is also often found in the literature. It has the probability density function

:$f(x;k,lambda,\; heta)=\{k\; over\; lambda\}\; left(\{x\; -\; heta\; over\; lambda\}\; ight)^\{k-1\}\; e^\{-(\{x-\; heta\; over\; lambda\})^k\},$

for $x\; geq\; heta$ and "f"("x"; "k", λ, θ) = 0 for "x" < θ, where $k\; >0$ is the shape parameter, $lambda\; >0$ is the scale parameter and $heta$ is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.

The cumulative distribution function for the 2-parameter Weibull is

:$F(x;k,lambda)\; =\; 1-\; e^\{-(x/lambda)^k\},$

for "x" ≥ 0, and "F"("x"; "k"; λ) = 0 for "x" < 0.

The cumulative distribution function for the 3-parameter Weibull is

:$F(x;k,lambda,\; heta)\; =\; 1-\; e^\{-(\{x-\; heta\; over\; lambda\})^k\}$

for "x" ≥ θ, and "F"("x"; "k", λ, θ) = 0 for "x" < θ.

The failure rate "h" (or hazard rate) is given by

:$h(x;k,lambda)\; =\; \{k\; over\; lambda\}\; left(\{x\; over\; lambda\}\; ight)^\{k-1\}.$

Information entropy

The information entropy is given by

:$H=gammaleft(1!-!frac\{1\}\{k\}\; ight)+lnleft(frac\{lambda\}\{k\}\; ight)+1$

where $gamma$ is the Euler–Mascheroni constant.

Generating Weibull-distributed random variates

Given a random variate "U" drawn from the uniform distribution in the interval (0, 1), then the variate

:$X=lambda\; (-ln(U))^\{1/k\},$

has a Weibull distribution with parameters "k" and λ. This follows from the form of the cumulative distribution function. Note that if you are generating random numbers belonging to (0,1), exclude zero values to avoid the natural log of zero.

Related distributions

*$X\; sim\; mathrm\{Exponential\}(lambda)$ is an exponential distribution if $X\; sim\; mathrm\{Weibull\}(k\; =\; 1,\; lambda^\{-1\})$.
* is a Rayleigh distribution if .
*$lambda(-ln(X))^\{1/k\},$ is a Weibull distribution if $X\; sim\; mathrm\{Uniform\}(0,1)$.
* Inverse Weibull distribution with p.d.f. $f(x;k,lambda)=(k/lambda)\; (lambda/x)^\{(k+1)\}\; e^\{-(lambda/x)^k\}$
* See also the generalized extreme value distribution.

Uses

The Weibull distribution is used
* In survival analysis
* To represent manufacturing and delivery times in industrial engineering
* In extreme value theory
* In weather forecasting
* In reliability engineering and failure analysis (the most common usageFact|date=April 2008)
* In radar systems to model the dispersion of the received signals level produced by some types of clutters
* To model fading channels in wireless communications, as the Weibull fading model seems to exhibit good fit to experimental fading channel measurements
* In General insurance to model the size of Reinsurance claims, and the cumulative development of Asbestosis losses
* In forecasting technological change (also known as the Sharif-Islam model)
* To describe wind speed distributions, as the natural distribution often matches the Weibull shape

The Weibull distribution may be used in place of the normal distribution because a Weibull variate can be generated through inversion. Normal variates are typically generated using the more complicated Box-Muller method, which requires two uniform random variates.

The 2-Parameter Weibull distribution is used to describe the particle size distribution of particles generated by grinding, milling and crushing operations. The Rosin-Rammler distribution predicts fewer fine particles than the Log-normal distribution. It is generally most accurate for narrow PSDs.

Using the cumulative distribution function:
* "F(x; k; λ)" is the mass fraction of particles with diameter < "x"
* "λ" is the mean particle size
* "k" is a measure of particle size spread

References

Bibliography

*cite web |url=http://www.erpt.org/014Q/nelsa-06.htm |title=Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution |accessdate=2008-02-05 |last=Nelson, Jr |first=Ralph |date=2008-02-05

External links

* [http://www.barringer1.com/wa_files/Weibull-ASME-Paper-1951.pdf A Statistical Distribution Function of Wide Applicability (the original 1951 article).]
* [http://www.xycoon.com/Weibull.htm The Weibull distribution (with examples, properties, and calculators).]
* [http://www.itl.nist.gov/div898/handbook/eda/section3/weibplot.htm The Weibull plot.]
* [http://www.weibull.com/GPaper/index.htm Weibull plotting paper.]
* [http://www.mathpages.com/home/kmath122/kmath122.htm Mathpages - Weibull Analysis]
* [http://www.qualitydigest.com/jan99/html/weibull.html Using Excel for Weibull Analysis]
This article from the Quality Digest describes how to use MS Excel to analyse lifetest data with the Weibull statistical distribution. Although Excel doesn't have an inverse Weibull function, this article shows how to use Excel to solve for critical values.
* [http://www.bobabernethy.com/bios_weibull.htm Biography of Waloddi Weibull. ]
* The SOCR Resource provides [http://socr.ucla.edu/htmls/SOCR_Distributions.html interactive interface to Weibull distribution] .