Probability density function Weibull

In association with each invariant, the a values correspond to the Weibull shape parameters and the /3 values correspond to Weibull scale parameters. A two-parameter Weibull probability density function has the following form... 

The mechanical performance of femoral heads and acetabular cups made from alumina have been the subject of intense research and development effort as it is crucial for the longevity of the endoprosthetic hip implant. Hence, the failure probability of these ceramic construction parts has been investigated and expressed by the Weibull probability density function. As safe design of... 

Let s assume that the / is the Weibull probability density function... 

In order to generalize the noise 200 samples of the noises are taken and analyzed with a statistical tool. The amplitudes obtained are modeled by a Weibull Probability Density Function (PDF), as it is suggested for similar noises on ADSL analysis [15,16]. 

Reliasoft Corporation, Weibull Probability Density Function, 1996-2006, available online at http //www.weibull.com/LifeDataWeb/weibull probability density function.htm. 

Establishing Mean Life to Failure. The mean cycles are determined from fitting the data at each stress level to a two-parameter Weibull distribution and computing the mean from the cumulative distribution equation for a probability of 50%. The two-parameter Weibull probability density function and cumulative distribution function as given by Madayag (1969) are... 

Fig. 4.12 Weibull probability density functions p(x) plotted for different m values...

The effects of scale parameter, r, on a Weibull probability density function curve are illustrated in Figure 6.12. A change in the scale parameter, r, has the same effect on the Weibull probability density function profile as changing the abscissa scale. The scale parameter also bears the same unit as the abscissa scale, in terms of either time or cycle. With a constant P value, when T] increases, the profile will flatten out, and the mean will move to the right... 

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

One has simply to assume a particular probability distribution for A with the survival function available in a closed form, namely the exponential, Erlang, Rayleigh, and Weibull. Table 9.1 summarizes the probability density functions, survival functions, and hazard rates for the above-mentioned distributions. In these expressions, A is the scale parameter and p and v are shape parameters with k, A, p > 0 and v = 1, 2,.... ... 

Probability density functions (pdfs) of transition time towards degraded states have been assumed to be Weibull functions, whereas the pdfs of failure times have been assumed to be ejq)onential. According to these hypotheses, the component failure rates depend only on the actual component degradation state and not directly on time, whereas transition rates between degraded states are time-dependent. 

In the first instance, when the results were analyzed by simple mean and standard deviation analysis, Amico et al. [16-18] got large relative standard deviatiOTi, indicating limitatimi of this method for the proper characterizatiOTi of the diameter. Then, they used Weibull probability density and cumulative distribution functions [20,56,58] to estimate two parameters, the characteristic life and a dimensionless positive pure number, which were supposed to determine the shape and scale of the distribution curve. For this, they adopted two methods, the maximum likelihood technique, which requires the solution of two nonlinear equations, and the analytical method using the probability plot as mentioned earlier for coir fibers. 

Where /(t) stands for probability density function, /(t) >0. From this it is clear that it is a case of three-parameter model. Shape and locations can be varied in accordance with selection of range of parameter for various variables viz. rj (>0), defines the bulk of the distribution parameter (scale), 8 (>0), determines shape (also often called Weibull slope) and y, defines the location of the distribution in time and has wide variation. 

The Weibull distribution adequately represents (Li et al., 2014, Liu, 2012) the wind speed probability distribution for most sampling times. Wind speed V is a variable generated at random (for a given time interval of one hour) with a Weibull distribution given by the following Probability Density Function (PDF) ... 

Figure 1. Histogram of failure data and probability density function (of Weibull distribution).

BM was applied to calculate posterior probability density functions of the parameters of Weibull distribution (formula (3)), Bayesian point esti-... 

Statistical data analysis of operation time till failure shows that operation time till failure T as random variable follows Weibull distribution (according to performed goodness of fit tests). The parameters k and p are assumed as independent random variables with prior probability density functions p x)—gamma pdf with mean value equals to prior (DPSIA) estimate of k and variance—10% of estimate value, / 2(j)— inverse gamma (as conjugate prior (Bernardo et al, 2003 Berthold et al, 2003)) pdf with mean value equals to prior (DPSIA) estimate of p and variance—10% of estimate value. Failure data tj, j =1,2,. .., 28. Thus, likelihood function is... 

In summary, the results of the second VDW-study make clear that it is possible to quantify existing safety-related functions of machine tools with a proven in use approach. The calculated maximum of the probability density function of the Weibull distribution is a useable value that can be compared with the quantitative requirements of the EN ISO 13849-1. 

This continuous random variable probability distribution was developed in the early 1950s by Walliodi Weibull, a Swedish professor in mechanical engineering [15]. The probability density function for the distribution is defined by... 

The Weibull distribution is more flexible than the exponential for these purposes. The probability density function for Weibull random variable is ... 

The functions wblpdf(x,a,b) and wblcdf(x,a,b) are available in the MATLAB Statistics Toolbox for calculating the probability density function and cumulative distribution function of Weibull. 

In all previous subsections the problem of detennining the parameters characteristic of a population of data or of a sample of data has been addressed with the use of a bell shaped distribution or normal distribution. Not always experimental data are effectively distributed in a symmetrical fashion. This, for instance, is the case of fatigue life at stress amplitudes close to fatigue Umit. A more general distribution function was introduced in 1939 by a Swedish engineer and researcher Weibull [7]. His probability density function in its simplest form (two parameters function) is defined as... 

Earthquake Recurrence, Fig. 2 Probability density functions and hazard rate functions for the Poisson model (panel a), the Weibull model (panel b), the gamma model (panel c), and the lognormal model... 

As a consequence, since the work of Bufe et al. (1977), a suite of models able to account for earthquake interaction and clustering has been introduced. We have presented those most used in practical applications such as Weibull, gamma, and lognormal and described their main features and differences in terms of their probability density functions and hazard functions. In addition we described a more physical model, that is, the Brownian passage time, which is currently used to perform time-dependent seismic hazard analysis in several areas, tectonic and volcanic areas such as California and Italy. 

Probability density function (Weibull distribution) of SWNT sub-bundle strength using/77 = 1.71 ando-Q = 17.8GPa. 

As an example of the type of functions possible with just one of the functions, Figure 8.12 shows various Weibull probability density plots for a range of Weibull shape parameters. The ability of this rather simple function to take on many different functional forms is one of the reasons it is frequently used for modeling of reliability data. The shape can range from an exponential like decrease with x to approximate a Gaussian for a shape parameter around 4 and an even more sharply... 

An example of the use of the derivative function to evaluate the probability density function is shown in Listing 8.4 for the Weibull function. This code in fact generates and saves the data used in plotting Figure 8.12. This code also gives pop-up plots of both the distribution function and the density function over a range of function parameters which in this case is the shape parameter of die Weibull function. The reader can view families of other distributions and probability densities by modifying the code on lines 8 and 16 for any of the other distribution function defined in this section. 

Listing 8.4. Illustration of evaluating the probability density function from derivative of a distribution funetion. Code also gives plots of Weibull distribution and density functions. 

Figure 12.13 Bimodal flaw size distribution a narrow peaked flaw population is superposed to a wide population, (a) Relative frequency of flaw sizes (bottom) and density of critical flaw sizes (top) versus the flaw size (b) Weibull plot showing the probability function (through line)...

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