Weibull Distribution

One of the most widely used distributions in reliability engineering is “Weibull Distribution“. It is a kind of versatile distribution that can take the values from the other distributions using the parameter called the shape parameter. Also, learn Probability Distribution here.

In this article, let us discuss the definition of Weibull distribution, formulas, properties, reliability, examples, two-parameter and the inverse Weibull distribution in detail.

Table of Contents:

Weibull Distribution Definition

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. It is an extreme value of probability distribution which is frequently used to model the reliability, survival, wind speeds and other data.

The only reason to use Weibull distribution is because of its flexibility. Because it can simulate various distributions like normal and exponential distributions. Weibull’s distribution reliability is measured with the help of parameters. 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 μ=0 and α =1the 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 parameters 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 γ < 1, then the failure rate decreases with time
  • If γ = 1, then the failure rate is constant

If γ > 1, the failure rate increases with time

Weibull Plot

The fit of Weibull distribution to data can be visually assessed using a Weibull plot. In other words, it is a graphical method for showing if a data set originates from a population that would inevitably be fit by a two-parameter Weibull distribution where the location is expected to be zero. This plot has unique scales that are designed so that if the data do support a Weibull distribution, the points will be linear or approximately linear. The least-squares fit of this line generates estimates for the shape and scale parameters of the Weibull distribution. In this case, the location is assumed to be zero. Precisely, the scale parameter is the exponent of the intercept and the shape parameter is the reciprocal of the slope of the fitted line. Thus, the below figure is a Weibull plot of a two-parameter distribution

Weibull plot

Weibull Distribution Reliability

The Weibull distribution is mostly used in reliability analysis and life data analysis because of its ability to adapt to different situations. Depending upon the parameter values, this distribution is used to model the variety of behaviours for a particular function. The probability density function usually describes the distribution function. The parameters in the distribution control the shape, scale and location of the probability density function. Several methods are used to measure the reliability of the data. But the Weibull distribution method is one of the best methods to analyse life data.

Weibull Distribution Properties

Some properties of Weibull distribution are:

  • Probability density function
  • Cumulative distribution function
  • Moments
  • Moment generating function
  • Shannon entropy

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

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

Examples

One such example of Weibull distribution is a 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 the Weibull analysis process is done by statistical software programs.

Keep visiting BYJU’S to get interesting articles and related videos to understand the concept in an easy and engaging way.

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