Weibull Distribution

The Weibull Distribution is a continuous probability distribution used to analyse life data,model failure times and access product reliability. It can also fit a huge range of data from many other fields like economics, hydrology, biology, engineering sciences.The two versions of Weibull probability density function(pdf) are

  • Two parameter pdf
  • Three parameter pdf

Weibull Distribution Formulas

The formula general weibull Distribution for three parameter pdf is given as

\(f(x)=\frac{\gamma }{\alpha }\left ( \frac{(x-\mu )}{\alpha } \right )^{{\gamma -1}}exp^{(-(\frac{(x-\mu )}{\alpha })^{\gamma })} \ \ \ \,x\geq \mu ;\gamma ,\alpha >0\)

Where,

  • \(\gamma\) is the shape parameter, also called as the Weibull slope or the threshold parameter.
  • \(\alpha\) is the scale parameter, also called the characteristic life parameter.
  • \(\mu\) is the location parameter, also called the waiting time parameter or sometimes the shift parameter.

The standard weibull distribution is derived, When \(\mu\)=0 and \(\alpha\)=1,the formula is reduced and it becomes

\(f(x)=\gamma x^{\gamma -1}exp^{(-x)^{\gamma }},x\geq 0;y> 0\)

Two Parameter Weibull distribution

The formula is practically similar to the three parameter Weibull, except that μ isn’t included:

\(f(x)=\frac{\gamma }{\alpha }\left ( \frac{(x)}{\alpha } \right )^{{\gamma -1}}exp^{(-(\frac{(x)}{\alpha })^{\gamma })} \ \ \ \,x\geq 0\)

The failure rate is determined by the value of the shape parameter \(\gamma\)

  • If \(\gamma < 1\), then the failure rate decreases with time
  • If \(\gamma = 1\), then the failure rate is constant
  • If \(\gamma > 1\), the failure rate increases with time

Weibull Distribution Example

One such example of Weibull distribution is weibull analysis which is used to study life data analysis(helps to measure time to failure rate). For example, Weibull analysis can be used to study:

  • Warranty Analysis
  • Components produced in a factory (like bearings, capacitors, or dielectrics),
  • Utility Services
  • Analyse the lifetime of dental and medical implants
  • Other areas where time-to-failure is important.

The analysis is also applicable in the design stage and in-service time as well and it is not only limited to the production stage. Now, the techniques to perform Weibull analysis process is done by statistical software programs.

Inverse Weibull Distribution

The inverse Weibull distribution has the ability to model failures rates which are most important in the reliability and biological study areas. Like weibull distribution,a three parameter inverse Weibull distribution is introduced to study the density shapes and failure rate functions.

The probability density function of the inverse Weibull distribution is given as

\(f(x)=\gamma \alpha ^{\gamma }x^{-(\gamma +1)}exp\left [ -\left ( \frac{\alpha }{x} \right )^{\gamma } \right ]\)

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Practise This Question

Identify the missing digit in the number 2344_6, if the number is divisible by 4.