The Weibull Distribution

The Fidelis Group, LLC

March 6, 2014

1 Introduction

The Weibull distribution is the most widely used distribution within reliability engineering.Waloddi Weibull, Swedish physicist, introduced the family of Weibull distributions in 1939. In his1951 article, ”A Statistical Distribution Function of Wide Applicability” (J. of Applied Mechanics,vol. 18: 293-297) he discusses many application for the Weibull distribution. In short, the Weibulldistribution can model infant mortality or ware out components. Infant mortality implies thatthere is a greater chance of failure in the early life of a component. Where as with ware out, thechance of failure increases with time. This paper will discuss the 2 parameter Weibull distribution,both probability density (PDF) and cumulative distribution functions (CDF), and the associatedparameters. Graphs are used to illustrate the varying of these parameters while holding the otherfixed.

2 Parameters

The Parameters for a Weibull distribution define the shape or look of the distribution. Theparameters are as follows:

Eta(η) − scale factor or characteristic life parameter

Beta(β) − shape factor or slope

The Titan equivalent parameters are

P1 = Eta(η)

P2 = Beta(β)

These parameters allow the Weibull distribution to model a wide range of diverse behaviors thatare time dependent. Below are descriptions of each parameter’s effects on the shape or look ofthe distribution.

1

2.1 Eta η

Eta, the scale factor or characteristic life, effects the height and width of the distribution. Eta(η) has the same units as does x. That is, days, hours, meters, ect. For example, suppose thatwe keep β constant, say β = 2. We then vary η, say η = 25; η = 50; η = 100; η = 200.

Figure 1: Weibull PDF Plots (varying η)The graph tothe right showsthe plots of thePDF. Note thatas we increaseη, the height de-creases and thewidth increases.This will in ef-fect stretch outthe distribution.In general, if β isfixed then:

if η is decreased : the height will increase and width will decrease

if η is increased : the height will decrease and width will increase

Figure 2: Weibull CDF Plots (varying η)The graph tothe right showsthe plots of theCDF. Note thatas we increase η,this stretches outthe CDF just aswe seen with thePDF. With re-spect to Titan,this increase therange of failureor repair timesthat Titan sam-ples. Notice thatwhen η = 50, thex− axis range isabout (0, 70). When η = 300 the x− axis range is about (0, 500).

2

2.2 Beta β

Beta, the shape factor, has the greatest effect on the distribution. Beta (β) is unit-less. Forexample, suppose that we keep η constant, say η = 100. We then vary β, say β = 0.5; β = 1;β = 3; β = 5.

Figure 3: Weibull PDF Plots (varying β)The graph tothe right showsthe plots of thePDF. Note thatas we change βthe concavity changes.This will effectthe overall shapeof the distribu-tion.In general, forany η > 0 wehave:

if 0 < β < 1 : Concave Upward - Infant mortality

if β = 1 : Equal to the Exponential Distribution

if β > 1 : Concave Downward - Failure Rate Increases with x

Figure 4: Weibull CDF Plots (varying β)The graph tothe right showsthe plots of theCDF. Note thatall of the graphsintersect atF (η) = 0.6321.That is, were 63.21%of failures occured.This is an effectof CDF.Given η, β thenevaluate F at x =η. That is,

F (η; η, β) = 1 − e−( ηη )β

= 1 − e−(1)β

= 1 − e−1

≈ 0.6321

3

Hence, when η is fixed, all CDFs (when β is varied) will intersect at the same point.

3 Probability Distribution Function (PDF)

Definition 1 Let X be a random variable and let η, β > 0, then X is said to have a WeibullDistribution with parameters η and β if the PDF of X is

f(x; η, β) =

βη

(xη

)β−1

e−( xη )β

, x ≥ 0

0, else(1)

4 Cumulative Distribution Function (CDF)

Definition 2 Let X be a random variable and let η, β > 0, then X is said to have a WeibullDistribution with parameters η and β if the CDF of X is

F (x; η, β) =

{1 − e−( xη )

β

, x ≥ 0

0, else(2)

5 Mean, Variance, Standard Deviation

Definition 3 Let X be a random variable such that it is Weilbull distributed with parametersη, β > 0, then the Mean of X is defined by

E(X) = ηΓ

(1 +

1

β

), (3)

where Γ(∗) is the gamma function.

Definition 4 Let X be a random variable such that it is Weilbull distributed with parametersη, β > 0, then the Variance of X is defined by

V ar(X) = η2

[Γ

(1 +

2

β

)−(

Γ

(1 +

1

β

))2], (4)

where Γ(∗) is the gamma function.

Definition 5 Let X be a random variable such that it is Weilbull distributed with parametersη, β > 0, then the Standard Deviation of X is defined by

StDev(X) =√V ar(X), (5)

where V ar(X) is definded by (4).

4