韦伯分布.docx

1、韦伯分布韋伯分佈韋伯分佈(Weibull distribution)以指數分佈為一特例。其p.d.f.為其中,0。以表此分佈, 有二參數, 為尺度參數, 為形狀參數。若取=1, 則得分佈, 以表之。底下給出一些韋伯分佈p.d.f.之圖形。韋伯分佈是瑞典物理學家Waloddi Weibull, 為發展強化材料的理論, 於西元1939年所引進, 是一較新的分佈。在可靠度理論及有關壽命檢定問題裡, 常少不了韋伯分佈的影子。分佈的分佈函數為 期望值與變異數分別為Characteristic Effects of the Shape Parameter, , for the Weibull Distri

2、butionThe Weibull shape parameter, , is also known as the slope. This is because the value of 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 para

3、meter will cause the distribution equations to reduce to those of other distributions. For example, when = 1, the pdf of the three-parameter Weibull reduces to that of the two-parameter exponential distribution or:where failure rate.The parameter is a pure number, i.e. it is dimensionless.The Effect

4、 of on the pdfFigure 6-1 shows the effect of different values of the shape parameter, , on the shape of the pdf. One can see that the shape of the pdf can take on a variety of forms based on the value of .Figure 6-1: The effect of the Weibull shape parameter on the pdf.For 0 1: f(T) = 0 at T = 0 (or

5、 ). f(T) increases as (the mode) and decreases thereafter. For 2.6 the Weibull pdf is positively skewed (has a right tail), for 2.6 3.7 it is negatively skewed (left tail).The way the value of relates to the physical behavior of the items being modeled becomes more apparent when we observe how its d

6、ifferent values affect the reliability and failure rate functions. Note that for = 0.999, f(0) = , but for = 1.001, f(0) = 0. This abrupt shift is what complicates MLE estimation when is close to one.The Effect of on the cdf and Reliability FunctionFigure 6-2: Effect of on the cdf on a Weibull proba

7、bility plot with a fixed value of .Figure 6-2 shows the effect of the value of 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 . Figur

8、e 6-3 shows the effects of these varied values of on the reliability plot, which is a linear analog of the probability plot.Figure 6-3: The effect of values of on the Weibull reliability plot. R(T) decreases sharply and monotonically for 0 1 and is convex. For = 1, R(T) decreases monotonically but l

9、ess sharply than for 0 1, R(T) decreases as T increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.The Effect of on the Weibull Failure Rate FunctionThe value of has a marked effect on the failure rate of the Weibull distribution and inferences can be draw

10、n about a populations failure characteristics just by considering whether the value of is less than, equal to, or greater than one.Figure 6-4: The effect of on the Weibull failure rate function.As indicated by Figure 6-4, populations with 1 have a failure rate that increases with time. All three lif

11、e stages of the bathtub curve can be modeled with the Weibull distribution and varying values of .The Weibull failure rate for 0 1, (T) increases as T increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1 2, the (T) curve is concave, conse

12、quently the failure rate increases at a decreasing rate as T increases.For = 2 there emerges a straight line relationship between (T) and T, starting at a value of (T) = 0 at T = , and increasing thereafter with a slope of . Consequently, the failure rate increases at a constant rate as T increases. Furthermore, if = 1 the slo