
Characteristics of the Weibull Distribution
In the last several issues of Reliability HotWire, we looked at how distributions are defined and how common reliability metrics are derived. In this issue, we will take a closer look at a specific distribution that is widely used in life data analysis  the Weibull distribution. Named for its inventor, Waloddi Weibull, this distribution is widely used in reliability engineering and elsewhere due to its versatility and relative simplicity.
As was discussed in February's Reliability Basics, a distribution is mathematically defined by its pdf equation. The most general expression of the Weibull pdf is given by the threeparameter Weibull distribution expression, or:
Where:
and:
 β is the shape parameter, also known as the Weibull slope
 η is the scale parameter
 γ is the location 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 twoparameter Weibull distribution. There is also a form of the Weibull distribution known as the oneparameter Weibull distribution. This in fact takes the same form as the twoparameter Weibull pdf, the only difference being that the value of β is assumed to be known beforehand. This assumption means that only the scale parameter needs be estimated, allowing for analysis of small data sets. It is recommended that the analyst have a very good and justifiable estimate for β before using the oneparameter Weibull distribution for analysis.
As was mentioned previously, the Weibull 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 behaviors. 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. This is 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 behavior 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 threeparameter Weibull reduces to that of the twoparameter exponential distribution. The parameter β is a pure number (i.e., it is dimensionless).
The following figure 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.
Looking at the same information on a Weibull probability plot, one can easily understand why the Weibull shape parameter is also known as the slope. The following plot shows how the slope of the Weibull probability plot changes with β. Note that the models represented by the three lines all have the same value of η.
Another characteristic of the distribution where the value of β has a distinct effect is the failure rate. The following plot shows the effect of the value of β on the Weibull failure rate.
This is one of the most important aspects of 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 earlylife 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 wearout failures. These comprise the three sections of the classic "bathtub curve." 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 that was identical to the bathtub curve. An example of a bathtub curve is shown in the following chart.
Weibull Scale parameter, η
A change in the scale parameter, η, has the same effect on the distribution as a change of the abscissa scale. Increasing 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 η is 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 η is 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.
 η has the same unit as T, such as hours, miles, cycles, actuations, etc.
Weibull Reliability Metrics
As was mentioned in last month's Reliability Basics, the pdf can be used to derive commonlyused reliability metrics such as the reliability function, failure rate, mean and median. The equations for these functions of the Weibull distribution will be presented in the following section, without derivations for the sake of brevity and simplicity. Note that in the rest of this section we will assume the most general form of the Weibull distribution, the threeparameter form. The appropriate substitutions to obtain the other forms, such as the twoparameter form where γ = 0, or the oneparameter form where β is a constant, can easily be made.
The Weibull reliability function is given by:
The Weibull failure rate function is given by:
The Weibull mean life, or MTTF, is given by:
where Γ(*) is the gamma function. The gamma function is defined as:
The equation for the median life, or B_{50} life, for the Weibull distribution is given by: