Template:Aw characteristics

Characteristics
The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, $$\beta ,$$  and eta,  $$\eta ,$$  and the effect they have on the  $$pdf,$$  reliability and failure rate functions.

Looking at $$\beta $$
Beta, $$\beta ,$$  is called the shape parameter or slope of the Weibull distribution. Changing the value of $$\beta $$  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 $$\beta $$  on the pdf
 * •	For $$0<\beta <1$$, the failure rate decreases with time and:
 * o	As $$T\to 0,$$   $$f(T)\to \infty .$$
 * o	As $$T\to \infty $$,  $$f(T)\to 0$$.
 * o	 $$f(T)$$ decreases monotonically and is convex as  $$T$$  increases.
 * o	The mode is non-existent.
 * •	For $$\beta =1,$$  it becomes the exponential distribution, as a special case,


 * or:


 * $$f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0$$


 * where $$\tfrac{1}{\eta }=\lambda =$$  chance, useful life, or failure rate.


 * •	For $$\beta >1$$,  $$f(T),$$  the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:
 * o	 $$f(T)=0$$ at  $$T=0$$.
 * o	 $$f(T)$$ increases as  $$T\to \tilde{T}$$  (mode) and decreases thereafter.
 * o	For $$\beta =2$$  it becomes the Rayleigh distribution as a special case. For  $$\beta <2.6$$  the Weibull  $$pdf$$  is positively skewed (has a right tail), for  $$2.6<\beta <3.7$$  its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal  $$pdf$$, and for  $$\beta >3.7$$  it is negatively skewed (left tail).
 * •	The parameter $$\beta $$  is a pure number, i.e. it is dimensionless.

Effects of $$\beta $$  on the Reliability Function and the cdf




 * •	 $$R(T)$$ decreases sharply and monotonically for  $$0<\beta <1$$, it is convex, and decreases less sharply for the same  $$\beta $$.
 * •	For $$\beta =1$$  and the same  $$\eta $$,  $$R(T)$$  decreases monotonically but less sharply than for  $$0<\beta <1$$ , and is convex.
 * •	For $$\beta >1$$,  $$R(T)$$  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 $$\beta $$  on the Failure Rate Function


 * •	The Weibull failure rate for $$0<\beta <1$$  is unbounded at  $$T=0$$ . The failure rate,  $$\lambda (T),$$  decreases thereafter monotonically and is convex, approaching the value of zero as  $$T\to \infty $$  or  $$\lambda (\infty )=0$$ . 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 $$\beta =1$$,  $$\lambda (T)$$  yields a constant value of $$\tfrac{1}{\eta }$$ , or:
 * $$\lambda (T)=\lambda =\frac{1}{\eta }$$

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
 * •	For $$\beta >1$$,  $$\lambda (T)$$  increases as  $$T$$  increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For  $$1<\beta <2$$  the  $$\lambda (T)$$  curve is concave, consequently the failure rate increases at a decreasing rate as  $$T$$  increases.
 * •	For $$\beta =2$$, or for the Rayleigh distribution case, the failure rate function is given by:
 * $$\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)$$

hence there emerges a straight line relationship between $$\lambda (T)$$  and  $$T$$, starting at a value of  $$\lambda (T)=0$$  at  $$T=0$$ , and increasing thereafter with a slope of  $$\tfrac{2}$$. Consequently, the failure rate increases at a constant rate as $$T$$  increases. Furthermore, if $$\eta =1$$  the slope becomes equal to 2, and  $$\lambda (T)$$  becomes a straight line which passes through the origin with a slope of 2.
 * •	When $$\beta >2$$  the  $$\lambda (T)$$  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 $$\eta $$
Eta, $$\eta ,$$  is called the scale parameter of the Weibull distribution. The parameter $$\eta $$  has the same units as  $$T$$, such as hours, miles, cycles, actuations, etc.
 * •	A change in the scale parameter $$\eta $$  has the same effect on the distribution as a change of the abscissa scale.
 * o	If $$\eta $$  is increased while  $$\beta $$  is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
 * o	If $$\eta $$  is decreased while  $$\beta $$  is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.