The Weibull Distribution

Fig. 18.3. (a) The Weibull distribution function, (b) When the modulus, m, changes, the survival probability changes os shown. 

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

Similarly, the strength parameters for the Weibull distribution ean be approximated by the same method to ... 

Repair and maintenance records were analyzed to determine failure rates and distribution of failure modes. Preliminary findings are reported which include the Weibull distribution characteristics. Failure mode distributions are approximate. Overall mean-time-between-failure is given for the kiln, leach tank, screwfeeder, tank pump, tank gearbox, and kiln gearbox. The study was confined to an analysis of unscheduled repairs and failures. 

Weibull distribution This distribution has been useful in a variety of reliability applications. The Weibull distribution is described by three parameters, and it can assume many shapes depending upon the values of the parameters. It can be used to model decreasing, increasing, and constant failure rates. 

Explain why a plant accident is more likely to happen during startup of a new plant or a retro-fit process. Refer to Chapter 20 and careful review the presentation or tlie bathtub curve tliat is represented by the Weibull distribution. 

The Weibull distribution provides a iiiatliematical model of all tliree stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tliree stages of tlie batlitub curve is... 

Equation (20.4.3) defines tlie pdf of the Weibull distribution. Tlie exponential distribution, whose pdf is given in Eq. (20.4.1), is a special case of the Weibull distribution witli p = 1. Tlie variety of assumptions about failure rate and tlie probability distribution of time to failure tliat can be accommodated by the Weibull distribution make it especially attractive in describing failure time distributions in industrial and process plant applications. 

As anollier example of probability calculations - lliis time involving the Weibull distribution - consider a component whose time to failure T in hours lias a Weibull pdf with parameters a = 0.01 and p = 0.50. To find llie probability that llie component will operate for at least 8100 hours, substitute a = 0,01 and p = 0.50 in Eq. (20.4.3), Tliis gives... 

The Weibull distribution will again find application in tlie case study reviewed in Section 21.6. 

For each of the 36 bus sections tliat had not already failed, the Weibull distribution was used to detennine tlie probability of failure before tlie next outage. Under assumption (a), tliis probability is P(T < 3301T > 209) i.e., tlie conditional probability of failure before 330 days, given tliat tlie bus section lias survived 209 days. Under assumption (b), tlie corresponding probability is P(T < 330 T > 230). For part (b), tlie estimates of the Weibull distribution parameters used in part (a) were modified to take into consideration tlie absence of failures for 3 additional weeks. 

Three ranges of values of n were considered, >1, 0.7—1.0 and <0.7. When n> 1, and particularly when 3 < n < 4, the Weibull distribution readily reduces to a normal distribution if the Erofe ev function is symmetrical about a = 0.5. [The Weibull distribution is symmetrical for n = 3.26, i.e. (1 — In 2)-1, and the inflection point varies only slowly with n.] Thus, under these conditions (3 < n < 4 and symmetry about a = 0.5), we may derive the parameters of the corresponding normal distribution (where p defines the half-life of the reaction and the dispersion parameter, a, is a measure of the lack of homogeneity of the surface centres), viz. 

When n < 0.7, the ln[—ln(l — a)] against In t plots show curvature and linearity is improved if the latter parameter is replaced by t. This reduces the Weibull distribution to a log-normal distribution. Since both exponential and normal distributions are special cases of the more general gamma distribution, Kolar-Anic and Veljkovic [441] compared the applicability of the Weibull and the gamma distributions. The shape parameter of the latter (e) was shown to depend exclusively on the shape parameter of the former (n). 

These various distributions reflect the variations in reactivity of the reacting sites. Johnson and Kotz [444] discuss in detail the Weibull and other distributions which find application when conditions of strict randomness of the exponential distribution are not satisfied. From an empirical point of view, the power transformation is a practical and convenient method of introducing a degree of flexibility into a model. Gittus [445] has discussed some situations in which the Weibull distribution may be expected to find application, including nucleation and growth processes in alloy transformations. 

Dorko et al. [442] have used the Weibull distribution function for the consideration of reactions in which decomposition is accompanied by melting. Following a procedure described by Kao [446], they used a mixed Weibull function, written as a linear combination of separate functions, viz. 

Although the Noyes-Whitney equation has been used widely, it has been shown to be inadequate in modeling either S-shape experimental data or data with a steeper initial slope. Therefore, a more general function, based on the Weibull distribution [8], was proposed [9] and applied empirically and successfully to all types of dissolution curves [10] ... 

F. Langenbucher. Linearization of dissolution rate curves by the Weibull distribution. J. Pharm. Pharmacol. 

Figure 5.9 gives the annual wind speed distribution of a site where the wind potential is exploitable. The wind speed values are sorted in the ascending order to be used in Equation 5.1. An alternative way to simulate the annual wind speed distribution is using the Weibull distribution with the proper pair of constants. 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

The Weibull distribution, illustrated in Figure 2, is most attractive, as it permits characterization of all typical cases of a PDF and CDF with only three parameters (2-4) ... 

An outstanding feature of the Weibull distribution is that it provides a clear separation of this parameter from the exponential part reflecting rate and shape of the profile. 

The mean represents the overall rate of the relevant process and corresponds to the abscissa of the center of gravity of the PDF and the mean value of the CDF. It is exactly reflected by the rate parameter of the Weibull distribution t63 2% is exact for mono-exponential and may be used as a shorthand estimate for any CDF of similar shape. 

According to Table 1, semi-invariants of higher order characterize the shape of the profile in terms of variance, skewness, and kurtosis. The outstanding merit of the Weibull distribution is that its shape parameter a provides a summarizing measure for this property. For other distributions, the characterization of the shape is less obvious. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

Validation was carried out using the area under the dissolution curve, this being the major response to be optimized. The dissolution curves fitted the Weibull distribution. 

Table 11 contains the pertinent parameter estimates and the residual error for each release model. From these results and from Figs 4, 5 and 6 it can be concluded that the release of diclofenac sodium is fitted by the Weibull distribution. P>1 (P being the shape parameter) is characteristic for a slower initial rate (diclofenac sodium is insoluble at pH 1.2) followed by an acceleration to the final plateau (sigmoid). In the direct compression optimization, after infinite time, the fraction released (F.rJ is estimated to be only 90% [13]. 

As expected, when compared with the direct compression results [13], double compression considerably enhanced the flow of the powder and substantially decreased the variability between tablets. The drug release followed the Weibull distribution. 

The following data were generated by the Weibull distribution of Exercise 17 ... 

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