1. Exploratory Data Analysis1.3. EDA Techniques1.3.6. Probability Distributions1.3.6.6. Gallery of Distributions

1.3.6.6.8. Weibull Distribution

ProbabilityDensityFunction

The formula for the probability density function of the generalWeibull distribution is

where γ is the shape parameter, μ is the location parameter andα is the scale parameter. The case where μ = 0 and α = 1 iscalled the standard Weibull distribution. The case where μ =0 is called the 2-parameter Weibull distribution. The equationfor the standard Weibull distribution reduces to

Since the general form of probability functions can beexpressed in terms of the standard distribution, all subsequentformulas in this section are given for the standard form of thefunction.

The following is the plot of the Weibull probability densityfunction.

CumulativeDistributionFunction

The formula for the cumulative distribution function of theWeibull distribution is

f (x) = ( exp (− ((x − μ)/ α ) x ≥ μ; γ, α > 0γα

x− μα )(γ− 1) )γ

f (x) = γ exp(− ( )) x ≥ 0; γ > 0x(γ− 1) xγ

F (x) = 1 − x ≥ 0; γ > 0e− ( )xγ

1.3.6.6.8. Weibull Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

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The following is the plot of the Weibull cumulative distributionfunction with the same values of γ as the pdf plots above.

PercentPointFunction

The formula for the percent point function of the Weibulldistribution is

The following is the plot of the Weibull percent point functionwith the same values of γ as the pdf plots above.

HazardFunction

The formula for the hazard function of the Weibull distributionis

The following is the plot of the Weibull hazard function withthe same values of γ as the pdf plots above.

G(p) = (− ln(1 − p) 0 ≤ p < 1; γ > 0)1/ γ

h(x) = γ x ≥ 0; γ > 0x(γ− 1)

1.3.6.6.8. Weibull Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

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CumulativeHazardFunction

The formula for the cumulative hazard function of the Weibulldistribution is

The following is the plot of the Weibull cumulative hazardfunction with the same values of γ as the pdf plots above.

SurvivalFunction

The formula for the survival function of the Weibulldistribution is

The following is the plot of the Weibull survival function withthe same values of γ as the pdf plots above.

H (x) = x ≥ 0; γ > 0xγ

S(x) = exp − ( ) x ≥ 0; γ > 0xγ

1.3.6.6.8. Weibull Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

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InverseSurvivalFunction

The formula for the inverse survival function of the Weibulldistribution is

The following is the plot of the Weibull inverse survivalfunction with the same values of γ as the pdf plots above.

CommonStatistics

The formulas below are with the location parameter equal tozero and the scale parameter equal to one.

Mean

where Γ is the gamma function

Median

Z (p) = (− ln(p) 0 ≤ p < 1; γ > 0)1/ γ

Γ( )γ+ 1γ

Γ(a) = dt∫ ∞0 ta− 1e− t

ln(2)1/ γ

1.3.6.6.8. Weibull Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

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Mode

Range 0 to .StandardDeviationCoefficient ofVariation

ParameterEstimation

Maximum likelihood estimation for the Weibull distribution isdiscussed in the Reliability chapter (Chapter 8). It is alsodiscussed in Chapter 21 of Johnson, Kotz, and Balakrishnan.

Comments The Weibull distribution is used extensively in reliabilityapplications to model failure times.

Software Most general purpose statistical software programs support atleast some of the probability functions for the Weibulldistribution.

(1 − γ > 11γ )1/ γ

0 γ ≤ 1∞

Γ( ) − (Γ( )γ+ 2γ

γ+ 1γ )2− −−−−−−−−−−−−−−

√

− 1Γ( )γ+ 2

γ

(Γ( )γ+ 1γ )2

− −−−−−−−−√

1.3.6.6.8. Weibull Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm

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