The Weibull Distribution

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} \,\! }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++ software.

Weibull Probability Density Function

The 3-Parameter Weibull

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Distribution Functions

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF) of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of:

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math]

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of: [math]\displaystyle{ \beta=1 \,\! }[/math] or:

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} \,\! }[/math]

The Median

The median, [math]\displaystyle{ \breve{T}\,\! }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} \,\! }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} \,\! }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} \,\! }[/math]

The Standard Deviation

The standard deviation, [math]\displaystyle{ \sigma _{T}\,\! }[/math], is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} \,\! }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} \,\! }[/math]

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} \,\! }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } \,\! }[/math]

These give the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R}\,\! }[/math], of a unit for a specified reliability, [math]\displaystyle{ R\,\! }[/math], starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} \,\! }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of [math]\displaystyle{ R\,\! }[/math]. If [math]\displaystyle{ R = 0.50\,\! }[/math], then [math]\displaystyle{ T_{R}=\breve{T} \,\! }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} \,\! }[/math]

Characteristics of the Weibull Distribution

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. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed 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 [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] 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 [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.