Characteristics of the 2-Parameter Weibull Distribution

The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta, β and eta, η, 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 Figure 3. In addition, when the cdf is plotted on Weibull probability paper, as shown in Figure 4, a change in beta is a change in the slope of the distribution on Weibull probability paper.

Effects of β on the pdf

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Fig. 3: Weibull pdf with 0 < β < 1, β = 1, β > 1 and a fixed η.

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where 1n.gif = λ = chance, useful life or failure rate.

Effects of β on the Reliability Function and the cdf

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Fig. 4: Weibull cdf, or unreliability vs. time, on Weibull probability plotting paper with 0 < β < 1, β = 1, β > 1 and a fixed η.

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Fig. 5: Weibull 1-cdf, or reliability vs. time, on linear scales with 0 < β < 1, β = 1, β > 1 and a fixed η.

Effects of β on the Failure Rate Function

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Fig. 6: Weibull failure rate vs. time with 0 < β < 1, β = 1, β > 1

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 T arrow.gif oo.gif or λ(oo.gif) = 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:

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

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 2n2.gif. 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.

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.

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Fig. 7: Weibull pdf with η = 50, η = 100, η = 200

See Also:
Weibull Distribution


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