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α,β>0。以表此分佈, 有二參數α,β, α為尺度參數, β為形狀參數。若取β=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, β, for the Weibull Distribution
The Weibull shape parameter, β, is also known as the slope. This is becau the value of β is equal to the slope of the regresd line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, som
e values of the shape parameter will cau the distribution equations to reduce to tho of other distributions. For example, when β = 1, the pdf of the three-parameter Weibull reduces to that of the two-parameter exponential distribution or:
where failure rate.
The parameter β is a pure number, i.e. it is dimensionless.
The Effect of β on the pdf
Figure 6-1 shows the effect of different values of the shape parameter, β, on the shape of the pdf. One can e that the shape of the pdf can take on a variety of forms bad on the value of β.
Figure 6-1: The effect of the Weibull shape parameter on the pdf.
For 0 < β 1:
∙ As (or γ),
∙ As , .
∙ f(T) decreas monotonically and is convex as T increas beyond the value of γ.
∙ The mode is non-existent.
For β > 1:
∙ f(T) = 0 at T = 0 (or γ).
∙ f(T) increas as (the mode) and decreas thereafter.
∙ 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). Conquently, it may approximate the normal pdf, and for β > 3.7 it is negatively skewed (left tail).
The way the value of β relates to the physical behavior of the items being modeled becomes more apparent when we obrve how its different values affect the reliability and failure rate functions. Note that for β = 0.999, f(0) = , but for β = 1.001, f(0) = 0. This abrupt shift is what complicates MLE estimation when β is clo to one.
The Effect of β on the cdf and Reliability Function
Figure 6-2: Effect of β on the cdf on a Weibull probability plot with a fixed value of η.
Figure 6-2 shows the effect of the value of β on the cdf, as manifested in the Weibull probability plot. It is easy to e why this parameter is sometimes referred to as the slope. Note that the models reprented by the three lines all have the same value of η. F
igure 6-3 shows the effects of the varied values of β on the reliability plot, which is a linear analog of the probability plot.
Figure 6-3: The effect of values of β on the Weibull reliability plot.
∙ R(T) decreas sharply and monotonically for 0 < β < 1 and is convex.
∙ For β = 1, R(T) decreas monotonically but less sharply than for 0 < β < 1 and is convex.
∙ For β > 1, R(T) decreas as T increas. As wear-out ts in, the curve goes through an inflection point and decreas sharply.
The Effect of β on the Weibull Failure Rate Function
The value of β has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of β is less than, equal to, or greater than one.
Figure 6-4: The effect of β on the Weibull failure rate function.
As indicated by Figure 6-4, populations with β < 1 exhibit a failure rate that decreas with time, populations with β = 1 have a constant failure rate (consistent with the exponential distribution) and populations with β > 1 have a failure rate that increas with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of β.
The Weibull failure rate for 0 < β < 1 is unbounded at T = 0 (or γ). The failure rate, λ(T), decreas thereafter monotonically and is convex, approaching the value of zero as or λ( ) = 0. This behavior makes it suitable for reprenting the failure rate of units exhibiting early-type failures, for which the failure rate decreas with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping.
For β = 1, λ(T) yields a constant value of or:
This makes it suitable for reprenting the failure rate of chance-type failures and the uful life period failure rate of units.
For β > 1, λ(T) increas as T increas and becomes suitable for reprenting the failure rate of units exhibiting wear-out type failures. For 1 < β < 2, the λ(T) curve is concave, conquently the failure rate increas at a decreasing rate as T increas.
