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Weibull distribution
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==Applications== The Weibull distribution is used{{Citation needed|date=June 2010}} [[File:FitWeibullDistr.tif|thumb|240px|Fitted cumulative Weibull distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]]<ref>{{cite web|url=https://www.waterlog.info/cumfreq.htm|title=CumFreq, Distribution fitting of probability, free software, cumulative frequency}}</ref> ]] [[File:DCA with four RDC.png|thumb|240px|Fitted curves for oil production time series data<ref name="ReferenceA">{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bani|last2=Mallick| title = Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas|journal=Sankhya B|year=2021|volume=84 |pages=1โ43 |doi=10.1007/s13571-020-00245-8|doi-access=free}}</ref>]] * In [[survival analysis]] * In [[reliability engineering]] and [[failure analysis]] * In [[electrical engineering]] to represent overvoltage occurring in an electrical system * In [[industrial engineering]] to represent [[manufacturing]] and [[Delivery (commerce)|delivery]] times * In [[extreme value theory]] * In [[weather forecasting]] and the [[Wind power#Wind energy resources|wind power industry]] to describe [[Wind power#Distribution of wind speed|wind speed distributions]], as the natural distribution often matches the Weibull shape<ref>{{cite web|url=http://www.reuk.co.uk/Wind-Speed-Distribution-Weibull.htm|title=Wind Speed Distribution Weibull โ REUK.co.uk|website=www.reuk.co.uk}}</ref> * In communications systems engineering ** In [[radar]] systems to model the dispersion of the received signals level produced by some types of clutters ** To model [[fading channel]]s in [[wireless]] communications, as the [[Weibull fading]] model seems to exhibit good fit to experimental fading [[Channel (communications)|channel]] measurements * In [[information retrieval]] to model dwell times on web pages.<ref>{{Cite book|last1=Liu|first1=Chao|last2=White|first2=Ryen W.|last3=Dumais|first3=Susan|date=2010-07-19|title=Understanding web browsing behaviors through Weibull analysis of dwell time|publisher=ACM|pages=379โ386|doi=10.1145/1835449.1835513|isbn=9781450301534|s2cid=12186028 }}</ref> * In [[general insurance]] to model the size of [[reinsurance]] claims, and the cumulative development of [[asbestosis]] losses * In forecasting technological change (also known as the Sharif-Islam model)<ref>{{cite journal |doi=10.1016/0040-1625(80)90026-8 |title=The Weibull distribution as a general model for forecasting technological change |journal=Technological Forecasting and Social Change |volume=18 |issue=3 |pages=247โ56 |year=1980 |last1=Sharif |first1=M.Nawaz |last2=Islam |first2=M.Nazrul }}</ref> * In [[hydrology]] the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. * In [[decline curve analysis]] to model oil production rate curve of shale oil wells.<ref name="ReferenceA"/> * In describing the size of [[Granular material|particles]] generated by grinding, [[mill (grinding)|milling]] and [[crusher|crushing]] operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the RosinโRammler distribution.<ref>[https://books.google.com/books?id=9RFdUPgpysEC&dq=Rosin-Rammler+distribution+2+Parameter+Weibull+distribution&pg=PA49 Computational Optimization of Internal Combustion Engine] page 49</ref> In this context it predicts fewer fine particles than the [[log-normal distribution]] and it is generally most accurate for narrow particle size distributions.<ref>{{cite book |last1=Austin |first1=L. G. |last2=Klimpel |first2=R. R. |last3=Luckie |first3=P. T. |title=Process Engineering of Size Reduction |date=1984 |publisher=Guinn Printing Inc. |location=Hoboken, NJ |isbn=0-89520-421-5}}</ref> The interpretation of the cumulative distribution function is that <math>F(x; k, \lambda)</math> is the [[Mass fraction (chemistry)|mass fraction]] of particles with diameter smaller than <math>x</math>, where <math>\lambda</math> is the mean particle size and <math>k</math> is a measure of the spread of particle sizes. * In describing random point clouds (such as the positions of particles in an ideal gas): the probability to find the nearest-neighbor particle at a distance <math>x</math> from a given particle is given by a Weibull distribution with <math>k=3</math> and <math>\rho=1/\lambda^3</math> equal to the density of the particles.<ref>{{cite journal |last=Chandrashekar |first=S. |title=Stochastic Problems in Physics and Astronomy |journal=Reviews of Modern Physics |volume=15 |number=1 |year=1943 |page= 86|doi=10.1103/RevModPhys.15.1 |bibcode=1943RvMP...15....1C }}</ref> * In calculating the rate of radiation-induced [[Radiation hardening#Digital_damage:_SEE|single event effects]] onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device [[Cross section (physics)|cross section probability]] data to a particle [[linear energy transfer]] spectrum.<ref>{{cite report|date= November 15, 2008|title= ECSS-E-ST-10-12C โ Methods for the calculation of radiation received and its effects, and a policy for design margins|url= https://ecss.nl/standard/ecss-e-st-10-12c-methods-for-the-calculation-of-radiation-received-and-its-effects-and-a-policy-for-design-margins/|publisher= European Cooperation for Space Standardization}}</ref> The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false{{cn|date=November 2023}} and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.<ref>{{cite report|author1=L. D. Edmonds|author2= C. E. Barnes|author3= L. Z. Scheick|date= May 2000|title= An Introduction to Space Radiation Effects on Microelectronics|url= https://parts.jpl.nasa.gov/pdf/JPL00-62.pdf|publisher= NASA Jet Propulsion Laboratory, California Institute of Technology|section= 8.3 Curve Fitting|pages=75โ76}}</ref>
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