Template:Aw characteristics

Characteristics

The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, [math]\displaystyle{ \beta , }[/math] and eta, [math]\displaystyle{ \eta , }[/math] and the effect they have on the [math]\displaystyle{ pdf, }[/math] reliability and failure rate functions.

Looking at [math]\displaystyle{ \beta }[/math]

Beta, [math]\displaystyle{ \beta , }[/math] is called the shape parameter or slope of the Weibull distribution. Changing the value of [math]\displaystyle{ \beta }[/math] forces a change in the shape of the [math]\displaystyle{ pdf }[/math] as shown in the next figure. In addition, when the [math]\displaystyle{ cdf }[/math] is plotted on Weibull probability paper, a change in beta is a change in the slope of the distribution on Weibull probability paper.
Effects of [math]\displaystyle{ \beta }[/math] on the pdf

Weibull pdf with [math]\displaystyle{ 0\lt \beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math] and a fixed [math]\displaystyle{ \eta }[/math].

• For [math]\displaystyle{ 0\lt \beta \lt 1 }[/math] , the failure rate decreases with time and:

o As [math]\displaystyle{ T\to 0, }[/math] [math]\displaystyle{ f(T)\to \infty . }[/math]

o As [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ f(T)\to 0 }[/math] .

o [math]\displaystyle{ f(T) }[/math] decreases monotonically and is convex as [math]\displaystyle{ T }[/math] increases.

o The mode is non-existent.

• For [math]\displaystyle{ \beta =1, }[/math] it becomes the exponential distribution, as a special case,

or:

[math]\displaystyle{ f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta \gt 0,T\ge 0 }[/math]

where [math]\displaystyle{ \tfrac{1}{\eta }=\lambda = }[/math] chance, useful life, or failure rate.

• For [math]\displaystyle{ \beta \gt 1 }[/math] , [math]\displaystyle{ f(T), }[/math] the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:

o [math]\displaystyle{ f(T)=0 }[/math] at [math]\displaystyle{ T=0 }[/math] .

o [math]\displaystyle{ f(T) }[/math] increases as [math]\displaystyle{ T\to \tilde{T} }[/math] (mode) and decreases thereafter.

o For [math]\displaystyle{ \beta =2 }[/math] it becomes the Rayleigh distribution as a special case. For [math]\displaystyle{ \beta \lt 2.6 }[/math] the Weibull [math]\displaystyle{ pdf }[/math] 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 [math]\displaystyle{ pdf }[/math] , and for [math]\displaystyle{ \beta \gt 3.7 }[/math] it is negatively skewed (left tail).

• The parameter [math]\displaystyle{ \beta }[/math] is a pure number, i.e. it is dimensionless.

Effects of [math]\displaystyle{ \beta }[/math] on the Reliability Function and the cdf

Weibull cdf, or Unreliability vs. Time, on Weibull probability plotting paper with [math]\displaystyle{ 0\lt \Beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math] and a fixed [math]\displaystyle{ \eta }[/math].

Weibull 1-cdf, or Reliability vs. Time, on linear scales with [math]\displaystyle{ 0\lt \Beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math] and a fixed [math]\displaystyle{ \eta }[/math].

• [math]\displaystyle{ R(T) }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0\lt \beta \lt 1 }[/math] , it is convex, and decreases less sharply for the same [math]\displaystyle{ \beta }[/math] .

• For [math]\displaystyle{ \beta =1 }[/math] and the same [math]\displaystyle{ \eta }[/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 [math]\displaystyle{ T }[/math] increases but less sharply than before, and as wear-out sets in, it decreases sharply and goes through an inflection point.

Effects of [math]\displaystyle{ \beta }[/math] on the Failure Rate Function

Weibull Failure Rate vs. Time with [math]\displaystyle{ 0\lt \Beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math], [math]\displaystyle{ \Beta\gt 1 }[/math].

• The Weibull failure rate for [math]\displaystyle{ 0\lt \beta \lt 1 }[/math] is unbounded at [math]\displaystyle{ T=0 }[/math] . The failure rate, [math]\displaystyle{ \lambda (T), }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ T\to \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 such behavior is encountered, one or more of the following conclusions can be drawn:

o Burn-in testing and/or environmental stress screening are not well implemented.

o There are problems in the production line.

o Inadequate quality control.

o Packaging and transit problems.

• For [math]\displaystyle{ \beta =1 }[/math] , [math]\displaystyle{ \lambda (T) }[/math] yields a constant value of [math]\displaystyle{ \tfrac{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] , or for the Rayleigh distribution case, the failure rate function is given by:

[math]\displaystyle{ \lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right) }[/math]

hence 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=0 }[/math] , and increasing thereafter with a slope of [math]\displaystyle{ \tfrac{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 [math]\displaystyle{ \lambda (T) }[/math] becomes a straight line which passes through the origin with a slope of 2.

• 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 wear-out life.

Looking at [math]\displaystyle{ \eta }[/math]

Eta, [math]\displaystyle{ \eta , }[/math] is called the scale parameter of the Weibull distribution. The parameter [math]\displaystyle{ \eta }[/math] has the same units as [math]\displaystyle{ T }[/math] , such as hours, miles, cycles, actuations, etc.

• A change in the scale parameter [math]\displaystyle{ \eta }[/math] has the same effect on the distribution as a change of the abscissa scale.

o If [math]\displaystyle{ \eta }[/math] is increased while [math]\displaystyle{ \beta }[/math] is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.

o If [math]\displaystyle{ \eta }[/math] is decreased while [math]\displaystyle{ \beta }[/math] is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.