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for and (;,,) = for <, where > is the shape parameter, > is the scale parameter and is the location parameter of the distribution. θ {\displaystyle \theta } value sets an initial failure-free time before the regular Weibull process begins.
It is characterized by a single parameter, λ, which is both the mean and variance of the distribution. The discrete Weibull distribution, on the other hand, is more flexible and can handle both over- and under-dispersion in count data. It has two parameters, q and β, which influence the shape and scale of the distribution.
Eq. 2 is the cumulative Weibull distribution with scale parameter and shape parameter ; = [^ ()] = constant factor depending on the structure geometry, = structure volume; = relative (size-independent) coordinate vectors, ^ = dimensionless stress field (dependent on geometry), scaled so that the maximum stress be 1; = number of spatial ...
In probability theory and statistics, a shape parameter (also known as form parameter) [1] is a kind of numerical parameter of a parametric family of probability distributions [2] that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter).
The Weibull modulus is a dimensionless parameter of the Weibull distribution. It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values.
In statistics, the exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape parameter. The cumulative distribution function for the exponentiated Weibull distribution is
Special cases of distributions where the scale parameter equals unity may be called "standard" under certain conditions. For example, if the location parameter equals zero and the scale parameter equals one, the normal distribution is known as the standard normal distribution, and the Cauchy distribution as the standard Cauchy distribution.
The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter σ {\displaystyle \sigma } is related to the Weibull scale parameter according to λ = σ 2 . {\displaystyle \lambda =\sigma {\sqrt {2}}.}