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Weibull

Discussion.

The Weibull distribution has two parameters, the 'shape' parameter and the 'scale' parameter. After you have given the parameters the SISA Weibull distribution program produces the mean and variance for the distribution you have given. If you additionally give an 'x' parameter the program returns the cumulative probability and the density value, if you give a cumulative probability value the program returns the appropriate 'x' value. The exponential distribution (used to study waiting times) is a special case of the Weibull distribution with alpha=1, mean=beta and lambda(the hazard rate)=1/beta. Another special case of the Weibull distribution is the Rayleigh distribution (used to study the scattering of radiation, wind speeds or to make certain transformations). For the Rayleigh distribution alpha is fixed at 2.

Random numbers from the Weibull distribution can be generated with the SISA Random page.

The alt-Weibull distribution has the same purpose as the usual Weibull distribution it is only 'slightly' differently shaped, according to the textbooks. It is in fact quite different. The alternative Weibull distribution is discussed by Mendenhall and Sincich (1995). You should have a sound reason to use this distribution.

In practice the Weibull distribution is used to describe two groups of phenomena. The lifetime of objects method is often used in quality control. A manufacturer provides the Weibull parameters for a product and the user can calculate the probability that a part fails after one, two, three or more years. The SISA Weibull distribution program allows you to do these calculations on the basis of already known parameters. For example, if you want to know the proportion that fails after one or more years, enter the value one in the 'x' box and read out the cumulative probability value. If you want to know the moment in time at which halve the parts will have failed, enter the value 0.5 in the '%' box and read out the 'x' value.

The description of wind speeds is an example of the use of the Weibull distribution to describe natural phenomena. Each part of the planet has it's own parameters for a Weibull distribution to describe the wind speed pattern in that place. On the basis of that you can calculate the number of days a year, or hours a day, with wind speeds above a certain force, or the mean wind speed, or the median wind speed, halve the days in a year have a wind speed below the median force, halve the days above. The Weibull distribution is very practical in this area because the distribution doesn't allow for negative value's and it is easy to appropriately consider the fact that on most days there will be a bit of wind, on some days a lot and you have those days that there is way too much wind speed.

The log-Weibull distribution concerns the log of a Weibull distributed random variable. It gives the limiting distribution for the smallest or largest values in samples drawn from a variety of distributions. The distribution is used to describe extreme conditions, such as extreme wind-gusts, extreme energy release during earthquakes, or extreme stress to which components are subjected. Sometimes the distribution is used as an alternative to the normal distribution in the case of skewed data. Other names for the log-Weibull are "Fisher-Tippett distribution" or "extreme value distribution". Although the distribution is the most used extreme value distribution there are other extreme value distributions describing the limiting distribution for the smallest or largest values drawn from particular distributions. Gumbel's distribution is a special case of the log Weibull distribution. For the Gumbel distribution alpha=0 and beta=1.

There are various statistical packages to estimate the Weibull parameters for a set of data. There are not so many packages for the log-Weibull. You will have to search the Internet for those. Unfortunately these packages tend to be expensive. SISA therefore might market a basic version of a program to estimate the Weibull distribution parameters soon.