# Weibull distribution 
In order to create a model of the failure distribution of a product, it is necessary to be able to model multiple types of failure: 

![Exemple de loi](baseUrl/fiabilite/weibull/Exemple-de-loi.png)


Therefore, considerable shape flexibility is necessary. The distribution primarily used for this is the Weibull distribution because it allows for a greater shape variability.

Because of its flexibility, the Weibull distribution makes the modelling of the behaviour of numerous types of failure possible, such as

- Components' resistance to rupture or the effort required to wear out metals. 
- The failure time of an electronic component. 
- The failure time for articles used outdoors such as pneumatic automobiles. 
- Systems that fail when the weakest component in the system contains a defect. 
The Weibull distribution also supports the modelling of the behaviour of different lifespan situations for one component. 
The function of the Weibull distribution is as follows:
![equation](R(t)=e^(-((t-del)/theta)^beta))

It is composed of 3 parameters: 
- β - shape parameter
- θ - scale parameter
- δ - delay parameter

## β, shape parameter :
It makes adaptation of the distribution's shape possible to place it as close as possible to the observed failure rate:

![Wiebull beta = 1](baseUrl/fiabilite/weibull/Wiebull-beta-1.png)

*<center>β = 1: The failure rate is constant (λ constant)</center>*


![wiebull beta = 2.6](baseUrl/fiabilite/weibull/wiebull-beta-2.6.png)


*<center>β > 1: The failure rate increases with time (λ increases - product's end of life)</center>*


![Weibull beta = 0.8](baseUrl/fiabilite/weibull/Weibull-beta-0.8.png)


*<center>β < 1: The failure rate decreases with time (λ decreases - infant failure)</center>*

## θ, scale parameter :
The θ parameter makes it possible to adjust the scale of the distribution law to the scale of the observed problem, for example:


![wiebull beta = 2.6](baseUrl/fiabilite/weibull/wiebull-beta-2.6.png)

Failure appears around t = 90.4

![Wiebull theta = 1000](baseUrl/fiabilite/weibull/Wiebull-theta-1000.png)

Failure appears around t = 904

## δ, delay parameter:
The δ parameter makes it possible to move the distribution law by one parameter δ

![wiebull beta = 2.6](baseUrl/fiabilite/weibull/wiebull-beta-2.6.png)

*<center>Β = 3,6 – θ = 100, δ = 0</center>*

Failure appears around t = 90.4

![Weibull delta = 100](baseUrl/fiabilite/weibull/Weibull-delta-100.png) 
*<center>Β = 3,6 – θ = 100, δ = 100</center>*

Failure appears around t = 190.4

Benutzerhandbuch

Weibull distribution

β, shape parameter :

θ, scale parameter :

δ, delay parameter:

Eine Komplettlösung für Industrie 4.0