Weibull distribution

The Weibull distribution is a continuous probability distribution over the set of positive real numbers. With a suitable choice of two parameters it resembles a normal distribution, an exponential distribution or other asymmetric distributions. For the modeling of the statistical distribution of wind speeds it is often used. Unlike a normal distribution into account the history of an object, it's memory afflicted. For example, aging, a component not only with the time, but depending on its use. The Weibull distribution describes the life and failure rate of electronic components or ( brittle ) materials. It can be adapted to increasing, constant and decreasing failure rates of technical systems. It is named after the Swedish engineer and mathematician Waloddi Weibull.

  • 2.1 Expectation value
  • 2.2 variance
  • 2.3 skewness
  • 2.4 entropy
  • 3.1 Weibullnetz
  • 3.2 Wind speed
  • 4.1 Relationship to the exponential distribution
  • 4.2 Stretched exponential

Definition

The Weibull distribution has two parameters.

Scale parameter

The scaling parameter, scale parameter or scaling factor.

In some applications, particularly in time-dependencies it is its inverse, replaces the characteristic lifetime. is on lifespan analyzes those time after which approximately 63.2 % of units have failed. This value is a parameter of the Weibull distribution.

If no scale parameter is specified, it is implicitly meant.

Shape parameter

The shape parameter or shape parameter or Weibull modulus is the parameter.

Alternatively, like the letters or used.

In practice, typical values ​​are in the range.

Due to the shape parameter, various specific probability distributions can be realized:

  • For there is the exponential distribution with a constant failure rate.
  • For there is the Rayleigh distribution.
  • For a distribution with vanishing skewness results (similar to the normal distribution ).

Density, distribution, etc.

Given a Weibull distribution with parameters.

The density function is:

The distribution function is:

The reliability or probability of survival is:

The failure rate is:

Properties

Expected value

Is the expected value of the Weibull distribution

With the gamma function.

Variance

Is the variance of the distribution

Skew

The skewness of the distribution

With the mean and standard deviation.

Entropy

The entropy of the Weibull distribution (expressed in nats ) is

The Euler - Mascheroni constant.

Applications

For systems with different causes of failure, such as technical components, these can be mapped with three Weibull distributions so that a can be represented " bathtub curve " .. The distributions then cover these three areas:

  • Early failures with, for example, in the run-in phase
  • Random failures with in the operating phase
  • Fatigue and wear failures at the end of product life with

In the mechanical process engineering, the Weibull distribution is applied as a specific particle size distribution. Here it is, however, called RRSB distribution ( Rosin, Rammler, Sperling and Bennet ).

Weibullnetz

Plotting the distribution in the form of

In a double- logarithmic diagram on which is also referred to as Weibullnetz, results in a straight line, in which we can easily read off the parameter as a slope. The characteristic lifetime can then be determined as follows:

Herein, the y-intercept.

It often happens that in spite of stress after an initial operating time failures occur ( for example, due to the wear of brake pads ). This can be taken into account in the Weibull distribution function. She then looks like this:

Plotting the function again, arises not a straight line but an upward convex curve. Shifts the value to all the points, the curve goes to a straight line.

Wind speed

The diagram shows a series of measurements of wind speeds (green). A Gaussian fit ( blue) numbers approaches insufficiently. Neither there is a negative wind speeds, there is the symmetrical distribution. A Weibull distribution introduces a second free parameter. Through them, the distribution of large and small wind speeds is very well approximated, as is the value to the maximum. From the fit parameters λ (1/5, 1 = 0.194 ) and k (2.00) followed by an expectation value of 4.5 m / s, in good agreement with the value of 4.6 m / s is determined from the measured values ​​.

Relationship to other distributions

Relationship to the exponential distribution

  • One sees that the case gives the exponential distribution. In other words, the exponential distribution deals with problems with constant failure rate. If we examine, however, problems with increasing () or decreasing ( ) failure rate, then you go from the exponential distribution on Weibull distribution.
  • Is the parameter, then a system is described having a rising time failure rate, that is an aging system.
  • Has an exponential distribution with parameter, then the random variable has a Weibull distribution. To prove this, consider the distribution function of:. Is the distribution function for a Weibull distribution.

Stretched exponential function

The function

Is referred to as a stretched exponential function.

Exponential decay#Mean lifetime Probability density function Entropy (information theory) Extreme value theory Beta-binomial distribution Boltzmann distribution Uniform distribution (discrete) Geometric distribution Logarithmic distribution Zeta distribution Triangular distribution Beta prime distribution Bose–Einstein statistics Erlang distribution F-distribution Fermi–Dirac statistics Generalized extreme value distribution Hotelling's T-squared distribution Cauchy distribution Gumbel distribution Stable distribution Logistic distribution Multinomial distribution Dirichlet distribution Multivariate normal distribution Wishart distribution
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