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 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 &lt; β &lt; 1, the failure rate decreases with time and:
 * As $$T\to 0,$$ $$f(T)\to \infty$$.
 * As $$T\to \infty $$, $$f(T)\to 0$$.
 * 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:


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


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


 * For β &gt; 1, f(T) , the Weibull distribution assumes wear-out type shapes (i.e., the failure rate increases with time) and:
 * f(T) = 0 at T = 0.
 * f(T) increases as $$T\to \tilde{T}$$ (mode) and decreases thereafter.
 * For β = 2 it becomes the Rayleigh distribution as a special case. For β &lt; 2.6, the Weibull pdf is positively skewed (has a right tail). For 2.6 &lt; β &lt; 3.7 , its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf , and for β &gt; 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 &lt; β &lt; 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 &lt; β &lt; 1 , and it is convex.
 * For β &gt; 1, R(T) decreases as T increases but less sharply than before. 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 &lt; β &lt; 1 is unbounded at T = 0 . The failure rate, λ(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:
 * Burn-in testing and/or environmental stress screening are not well implemented.
 * There are problems in the production line.
 * There is inadequate quality control.
 * There are packaging and transit problems.
 * For β = 1, λ(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 β &gt; 1, λ(T) increases as T increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1 &lt; β &lt; 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:
 * $$\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)$$

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 $$\tfrac{2}$$. 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 &gt; 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 η
Eta ( η ) is called the scale parameter of the Weibull distribution. The parameter η has the same units as T, such as hours, miles, cycles, actuations, etc.




 * A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale.
 * If η is increased while β is 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 is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.