# Weibull Distribution

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The Weibull distribution is a continuous probability distribution named after Swedish mathematician Waloddi Weibull. Nowadays, it’s commonly used to assess lifetime distribution of product reliability, analyze life data, profitability analysis and model failure times. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, ß (beta).

A distribution is mathematically defined by its pdf equation. The most general expression of the Weibull pdf is given by the three-parameter Weibull distribution expression i.e.

Where:

and:

• β is the shape parameter, also known as the Weibull slope or threshold parameter
• η is the scale parameter or the characteristic life parameter.
• γ is the location parameter or the waiting time parameter the shift parameter

Frequently, the location parameter is not used, and the value for this parameter can be set to zero. When this is the case, the pdf equation reduces to that of the two-parameter Weibull distribution.

In fact, when the value of ß is known before hand, i.e. when you just estimate the scale parameter- is is known as the the one-parameter Weibull distribution. (used for the analysis of small data sets)

This distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviours. An important aspect of the Weibull distribution is how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the pdf curve, the reliability and the failure rate.

Weibull Shape Parameter, β

The Weibull shape parameter, β, is also known as the Weibull slope, because the value of β is equal to the slope of the line in a probability plot. Different values of the shape parameter can have marked effects on the behaviour of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the pdf of the three-parameter Weibull reduces to that of the two-parameter exponential distribution. The parameter β is a pure number (i.e. it is dimensionless).

The figure on the left shows the effect of different values of the shape parameter, β, on the shape of the pdf (while keeping γ constant). One can see that the shape of the pdf can take on a variety of forms based on the value of β. While the figure on the right shows the probability plot which shows how the slope of the Weibull probability plot changes with the ß.

The different values of ß have also have a distinct effect on the failure rate as shown below:

As we study the effect of β on the Weibull distribution. As is indicated by the plot, Weibull distributions with β < 1 have a failure rate that decreases with time, also known as infantile or early-life failures. Weibull distributions with β close to or equal to 1 have a fairly constant failure rate, indicative of useful life or random failures. Weibull distributions with β > 1 have a failure rate that increases with time, also known as wear-out failures.

These effects can be well viewed together in a plot- A mixed Weibull distribution with one subpopulation with β < 1, one subpopulation with β = 1 and one subpopulation with β > 1 would have a failure rate plot.

Weibull Scale parameter, η

Increase in the value of η while holding β constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the “peak” of the pdf curve will also decrease with the increase of η, as indicated in the following figure.

• If ηis increased, while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If ηis decreased, while β and γ are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or γ), and its height increases.
• ηhas the same unit as T, such as hours, miles, cycles, actuations, etc.

We would also cover how Weibull distribution is used to determine the profitability curves of product in the future.