???? Weibull Distribution – Definition and Explanations

2023-06-22 03:38:13

Introduction

Weibull

Scale parameters (actual)
form (real) Support Probability density (In probability theory or in statistics, a probability density is…) (mass function) Distribution function (In probability theory or in statistics, the distribution function of a …) Median expectation (The term median, from the Latin medius, which is in the middle, has several acceptances in…) (center) Mode if k > 1 Variance ( In statistics and in probability, variance In thermodynamics, variance ) Asymmetry (Asymmetry is the absence of symmetry, or its inverse. In nature, crabs…) (statistics) Kurtosis
(non-normalized) see text Entropy (In thermodynamics, entropy is a state function introduced in 1865 by Rudolf Clausius…) Generating function (In mathematics, the generating function of the sequence (an) is the formal series defined by) times edit

In probability theory (Probability theory is the mathematical study of phenomena…), Weibull’s law, named following Waloddi Weibull, is a law of probability (In probability theory and in statistics, a law of probability described…) continues.

Characteristic functions

With two parameters its density (The density or relative density of a body is the ratio of its density to the…) probability (The probability (from the Latin probabilitas) is an assessment of the probable character of a… ) East :

where k > 0 is the shape parameter and λ > 0 the scale parameter of the distribution.

Its distribution function is defined by:

where, once more, x > 0.

With three parameters (generalized) its probability density is:

For x ≥ θ and f(x; k, λ, θ) = 0 for xk > 0 is the shape parameter, λ > 0 is the scale parameter and θ is the location parameter of the distribution.

Its distribution function for the 3-parameter Weibull distribution is defined by:

Pour x ≥ θ, et F(x; k, λ, θ) = 0 pour x

The coefficient (In mathematics a coefficient is a multiplicative factor which depends on a certain…) of asymmetry (skewness) is given by:

Kurtosis is given by:

The failure rate h is given by:

Convenient use

General

The expression Weibull’s law in fact covers a whole family of laws, some of them appearing in physics (Physics (from the Greek φυσις, nature) is etymologically the…) as a consequence of certain hypotheses. This is, in particular, the case of the exponential law (An exponential law corresponds to the following model:) (k = 1) and of Rayleigh’s law (In probability and in statistics, Rayleigh’s law is a law of probability to…) (k = 2) material (Matter is the substance that makes up any body having a tangible reality. Its…) of stochastic processes.

These laws above all constitute particularly useful approximations in various techniques, whereas it would be very difficult and without great interest to justify a particular form of law. A distribution with positive values ​​(or, more generally but less frequently, with values ​​greater than a given value) almost always looks the same. It starts from a frequency (In physics, frequency generally designates the measurement of the number of times a…) of zero appearance, increases to a maximum and decreases more slowly. It is then possible to find in the Weibull family a law which does not deviate too much from the data (In information technology (IT), a datum is an elementary description, often…) available by calculating and from the observed mean and variance.

Special application

The Weibull distribution (In probability theory, the Weibull distribution, named following Waloddi Weibull,…) is often used in the field of lifespan analysis (Life is the given name:) , thanks to its flexibility: as said previously, it makes it possible to represent at least approximately an infinity of probability distributions.

If the failure rate decreases over time (Time is a concept developed by human beings to understand…) then, k . If the failure rate is constant over time then k = 1. If the failure rate increases over time then k > 1. Understanding the failure rate can provide an indication (An indication (from the Latin indicare: to indicate ) is advice or a recommendation, written…) regarding the cause of failures.

A decreasing breakdown rate comes under “infant mortality”. Thus, the defective elements fail quickly, and the failure rate decreases over time, when the fragile elements leave the population. A constant failure rate suggests that the failures are related to a stationary cause. An increasing failure rate suggests “wear or a reliability problem (A system is reliable when the probability of fulfilling its mission over a period of time…)”: the elements have more and more chances of breaking down when the time spent.

It is said that the yield curve (In finance, the expression yield curve designates the graphical or mathematical representation in…) of breakdown is in the shape of a bathtub. Depending on the device, hip bath or swimming pool. Manufacturers and distributors have every interest in mastering this information by type of product in order to adapt:

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