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 β

Beta (β) is called the shape parameter or slope of the Weibull distribution. Changing the value of β forces a change in the shape of the pdf as shown in the next figure. In addition, when the cdf 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 β on the pdf

• For 0 < β < 1 , the failure rate decreases with time and:

• As [math]\displaystyle{ T\to 0, }[/math] [math]\displaystyle{ f(T)\to \infty . }[/math]
• As [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ f(T)\to 0 }[/math] .
• f(T) decreases monotonically and is convex as T increases.
• The mode is non-existent.

• For β = 1, 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 β > 1 , f(T), the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:

• f(T) = 0 at T = 0 .
• f(T) increases as [math]\displaystyle{ T\to \tilde{T} }[/math] (mode) and decreases thereafter.
• For β = 2 it becomes the Rayleigh distribution as a special case. For β < 2.6 the Weibull pdf is positively skewed (has a right tail), for 2.6 < β < 3.7 its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal <pdf, and for β > 3.7 it is negatively skewed (left tail).

• The parameter β is a pure number, i.e. it is dimensionless.

Effects of β on the Reliability Function and the cdf

• R(T) decreases sharply and monotonically for 0 < β < 1 , it is convex, and decreases less sharply for the same β .
• For β = 1 and the same η , R(T) decreases monotonically but less sharply than for 0 < β < 1 , and is convex.
• For β > 1 , R(T) decreases as T increases but less sharply than before, and as wear-out sets in, it decreases sharply and goes through an inflection point.

Effects of β on the Failure Rate Function

• The Weibull failure rate for 0 < β < 1 is unbounded at T = 0 . The failure rate, λ(T), 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:

• Burn-in testing and/or environmental stress screening are not well implemented.
• There are problems in the production line.
• Inadequate quality control.
• Packaging and transit problems.

• For β = 1 , λ(T) 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 β > 1 , λ(T) increases as T increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1 < β < 2 the λ(T) curve is concave, consequently the failure rate increases at a decreasing rate as T increases.
• For β = 2 , 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 λ(T) and T , starting at a value of λ(T) = 0 at T = 0 and increasing thereafter with a slope of [math]\displaystyle{ \tfrac{2}{{{\eta }^{2}}} }[/math] . Consequently, the failure rate increases at a constant rate as T increases. Furthermore, if η = 1 the slope becomes equal to 2, and λ(T) becomes a straight line which passes through the origin with a slope of 2.

• When β > 2 the λ(T) curve is convex, with its slope increasing as T increases. Consequently, the failure rate increases at an increasing rate as T 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.