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| {{More footnotes|date=June 2010}}
| | What are the greatest fad diets? It's a query I'm frequently asked. Unfortunately, I'm not really inside favour of fad diets, so it's a difficult 1 to answer.<br><br>So, again, back to the all-natural aspect; most individuals have found [http://safedietplans.com diets] effectively is changing the usual food with organic replacements. These are clearly marked; often not containing pesticides, artificial flavoring, or preservatives. Doctors might tell you that organic is the path to take, yet several persons create the mistake of thinking they could merely gorge themselves on organic food and everything is fine. That's no technique to follow, even diets that work.<br><br>You'll do better with healthy fat loss diet plans if you learn about correct portions. Check to create certain whatever program you choose is clear on part sizes. For instance, a medium apple may be the size of a tennis ball. A 3-ounce piece of meat may be the scale of a deck of cards.<br><br>Next, there is the meal delivery or prepackaged diet. This allows we to buy prepackaged meals. For instance should you purchase the three food and snacks for a day. This is very challenging, could be efficient for several period of time but could not be ideal for a extended expression objective considering we might not need to eat the prepackaged foods all the existence.<br><br>Thus, Fischer guarantees if you follow her dietary wellness plan over eight weeks, it is possible to consume plus sip your method into a modern, prettier hunting skin. She doesn't claim which the diets is a permanent plus miraculous remedy for genetic conditions like eczema. However, she does state which in the event you follow the principals of the diets that all of these conditions usually be vastly improved plus controlled.<br><br>The Cabbage Diet. Lose weight fast with the cabbage diet. This diet consists of creating a soup with cabbage, that we will eat throughout the 7-day length of the diet. The diet is low inside fat, and significant inside fiber which may enable you lose weight quickly. During this diet you will rotate between eating cabbage soup plus fruit with eating veggies and meat on every other day. This diet might aid we lose weight because it happens to be low inside calories. Some folks find it hard to keep this diet due to the limited food options it allows.<br><br>Learning that diets which function swiftly for females is a good initially step in lucrative fat reduction. Before we embark upon any diet plan it is best to recognize whether you should lose weight. BMI for ladies will enable you put into perspective your average weight thereby permitting you to better recognize whether a weight reduction program is right for you. |
| {{Probability distribution|
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| name =Weibull (2-Parameter)|
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| type =density|
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| pdf_image =[[Image:Weibull PDF.svg|325px|Probability distribution function]]<!--- NOT CORRECT (see discussion) [[Image:Weibul pdf.png|325px|Probability distribution function]]--->|
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| cdf_image =[[Image:Weibull CDF.svg|325px|Cumulative distribution function]]<!--- NOT CORRECT (see discussion) [[Image:Weibul cdf.png|325px|Cumulative distribution function]]--->|
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| parameters =<math>\lambda>0\,</math> [[scale parameter|scale]] ([[real number|real]])<br/><math>k>0\,</math> [[shape parameter|shape]] (real)|
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| support =<math>x \in [0; +\infty)\,</math>|
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| pdf =<math>f(x)=\begin{cases}
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| \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0\\
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| 0 & x<0\end{cases}</math>|
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| cdf =<math>\begin{cases}1- e^{-(x/\lambda)^k} & x\geq0\\ 0 & x<0\end{cases}</math>|
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| mean =<math>\lambda \, \Gamma(1+1/k)\,</math>|
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| median =<math>\lambda(\ln(2))^{1/k}\,</math>|
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| mode =<math>\begin{cases}
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| \lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\, &k>1\\
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| 0 &k=1\end{cases}</math>|
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| arg mode =<math>\lambda\frac{k-1}{k}^{\frac{1}{k}}\,</math> if <math>k>1</math>|
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| variance =<math>\lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,</math>|
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| skewness =<math>\frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}</math>|
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| kurtosis =(see text)|
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| entropy =<math>\gamma(1-1/k)+\ln(\lambda/k)+1 \,</math>|
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| mgf = <math>\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma(1+n/k), \ k\geq1</math>|
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| char = <math>\sum_{n=0}^\infty \frac{(it)^n\lambda^n}{n!}\Gamma(1+n/k)</math>
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| }}
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| In [[probability theory]] and [[statistics]], the '''Weibull distribution''' {{IPAc-en|ˈ|v|eɪ|b|ʊ|l}} is a continuous [[probability distribution]]. It is named after [[Waloddi Weibull]], who described it in detail in 1951, although it was first identified by {{harvtxt|Fréchet|1927}} and first applied by {{harvtxt|Rosin|Rammler|1933}} to describe a [[Particle-size distribution|particle size distribution]].
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| ==Definition==
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| The [[probability density function]] of a Weibull [[random variable]] is:<ref>Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes, 4th Edition</ref>
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| :<math>
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| f(x;\lambda,k) =
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| \begin{cases}
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| \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0 ,\\
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| 0 & x<0,
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| \end{cases}</math>
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| where ''k'' > 0 is the ''[[shape parameter]]'' and λ > 0 is the ''[[scale parameter]]'' of the distribution. Its [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|complementary cumulative distribution function]] is a [[stretched exponential function]]. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the [[exponential distribution]] (''k'' = 1) and the [[Rayleigh distribution]] (''k'' = 2).
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| If the quantity ''X'' is a "time-to-failure", the Weibull distribution gives a distribution for which the [[failure rate]] is proportional to a power of time. The ''shape'' parameter, ''k'', is that power plus one, and so this parameter can be interpreted directly as follows:
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| * A value of ''k'' < 1 indicates that the [[failure rate]] decreases over time. This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population.
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| * A value of ''k'' = 1 indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure.
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| * A value of ''k'' > 1 indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on.
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| In the field of [[materials science]], the shape parameter ''k'' of a distribution of strengths is known as the [[Weibull modulus]].
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| ==Properties==
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| ===Density function===
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| The form of the density function of the Weibull distribution changes drastically with the value of ''k''. For 0 < ''k'' < 1, the density function tends to ∞ as ''x'' approaches zero from above and is strictly decreasing. For ''k'' = 1, the density function tends to ''1/λ'' as ''x'' approaches zero from above and is strictly decreasing. For ''k'' > 1, the density function tends to zero as ''x'' approaches zero from above, increases until its mode and decreases after it. It is interesting to note that the density function has infinite negative slope at ''x'' = 0 if 0 < ''k'' < 1, infinite positive slope at ''x'' = 0 if 1 < ''k'' < 2 and null slope at ''x'' = 0 if ''k'' > 2. For ''k'' = 2 the density has a finite positive slope at ''x'' = 0. As ''k'' goes to infinity, the Weibull distribution converges to a [[Dirac delta distribution]] centred at ''x'' = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter.
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| ===Distribution function===
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| The [[cumulative distribution function]] for the Weibull distribution is
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| :<math>F(x;k,\lambda) = 1- e^{-(x/\lambda)^k}\,</math>
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| for ''x'' ≥ 0, and ''F''(''x''; ''k''; λ) = 0 for ''x'' < 0.
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| The quantile (inverse cumulative distribution) function for the Weibull distribution is
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| :<math>Q(p;k,\lambda) = \lambda {(-ln(1-p))}^{1/k}</math>
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| for 0 ≤ p < 1.
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| The [[failure rate]] ''h'' (or hazard rate) is given by
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| :<math> h(x;k,\lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k-1}.</math>
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| ===Moments===
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| The [[moment generating function]] of the [[logarithm]] of a Weibull distributed [[random variable]] is given by<ref name=JKB>{{harvnb|Johnson|Kotz|Balakrishnan|1994}}</ref>
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| :<math>E\left[e^{t\log X}\right] = \lambda^t\Gamma\left(\frac{t}{k}+1\right)</math>
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| where {{math|Γ}} is the [[gamma function]]. Similarly, the [[characteristic function (probability theory)|characteristic function]] of log ''X'' is given by
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| :<math>E\left[e^{it\log X}\right] = \lambda^{it}\Gamma\left(\frac{it}{k}+1\right).</math>
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| In particular, the ''n''th [[raw moment]] of ''X'' is given by
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| :<math>m_n = \lambda^n \Gamma\left(1+\frac{n}{k}\right).</math>
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| The [[mean]] and [[variance]] of a Weibull [[random variable]] can be expressed as
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| :<math>\mathrm{E}(X) = \lambda \Gamma\left(1+\frac{1}{k}\right)\,</math>
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| and
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| :<math>\textrm{var}(X) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,.</math>
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| The skewness is given by
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| :<math>\gamma_1=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}</math>
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| where the mean is denoted by {{math|μ}} and the standard deviation is denoted by {{math|σ}}.
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| The excess [[kurtosis]] is given by
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| :<math>\gamma_2=\frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2
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| -4\Gamma_1\Gamma_3+\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2}</math>
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| where <math>\Gamma_i=\Gamma(1+i/k)</math>. The kurtosis excess may also be written as:
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| :<math>\gamma_{2}=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}-3</math>
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| ===Moment generating function===
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| A variety of expressions are available for the moment generating function of ''X'' itself. As a [[power series]], since the raw moments are already known, one has
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| :<math>E\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right).</math>
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| Alternatively, one can attempt to deal directly with the integral
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| :<math>E\left[e^{tX}\right] = \int_0^\infty e^{tx} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}\,dx.</math>
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| If the parameter ''k'' is assumed to be a rational number, expressed as ''k'' = ''p''/''q'' where ''p'' and ''q'' are integers, then this integral can be evaluated analytically.<ref>See {{harv|Cheng|Tellambura|Beaulieu|2004}} for the case when ''k'' is an integer, and {{harv|Sagias|Karagiannidis|2005}} for the rational case.</ref> With ''t'' replaced by −''t'', one finds
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| :<math> E\left[e^{-tX}\right] = \frac1{ \lambda^k\, t^k} \, \frac{ p^k \, \sqrt{q/p}} {(\sqrt{2 \pi})^{q+p-2}} \, G_{p,q}^{\,q,p} \!\left( \left. \begin{matrix} \frac{1-k}{p}, \frac{2-k}{p}, \dots, \frac{p-k}{p} \\ \frac{0}{q}, \frac{1}{q}, \dots, \frac{q-1}{q} \end{matrix} \; \right| \, \frac {p^p} {\left( q \, \lambda^k \, t^k \right)^q} \right) </math>
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| where ''G'' is the [[Meijer G-function]].
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| The [[characteristic function (probability theory)|characteristic function]] has also been obtained by {{harvtxt|Muraleedharan|Rao|Kurup|Nair|2007}}.
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| ===Information entropy===
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| The [[information entropy]] is given by
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| :<math>
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| H(\lambda,k) = \gamma\left(1\!-\!\frac{1}{k}\right) + \ln\left(\frac{\lambda}{k}\right) + 1
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| </math>
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| where <math>\gamma</math> is the [[Euler–Mascheroni constant]].
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| ===Parameter estimation===
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| ====Maximum likelihood====
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| The [[maximum likelihood estimator]] for the <math>\lambda</math> parameter given <math>\beta</math> is,<ref>{{cite book
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| | author = Sornette, D.
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| | year = 2004
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| | title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>
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| :<math>\hat \lambda^k = \frac{1}{N} \sum_{i=1}^N (x_i^k - x_N^k)</math>
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| Where <math>x_1 > x_2 > ... > x_N</math> are the <math>N</math> largest observed samples.
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| The maximum likelihood estimator for <math>k</math> is,
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| :<math>
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| \hat k^{-1} = \frac{\sum_{i=1}^N (x_i^k \ln x_i - x_N^k \ln x_N)}
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| {\sum_{i=1}^N (x_i^k - x_N^k)}
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| - \frac{1}{N} \sum_{i=1}^N \ln x_i
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| </math>
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| This being an implicit function, one must generally solve for <math>k</math> by numerical means.
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| ==Weibull plot==
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| The fit of data to a Weibull distribution can be visually assessed using a Weibull Plot.<ref>[http://www.itl.nist.gov/div898/handbook/eda/section3/weibplot.htm The Weibull plot]</ref> The Weibull Plot is a plot of the [[empirical cumulative distribution function]] <math>\hat F(x)</math> of data on special axes in a type of [[Q-Q plot]]. The axes are <math>\ln(-\ln(1-\hat F(x)))</math> versus <math>\ln(x)</math>. The reason for this change of variables is the cumulative distribution function can be linearized:
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| :<math>\begin{align}
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| F(x) &= 1-e^{-(x/\lambda)^k}\\
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| -\ln(1-F(x)) &= (x/\lambda)^k\\
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| \underbrace{\ln(-\ln(1-F(x)))}_{\textrm{'y'}} &= \underbrace{k\ln x}_{\textrm{'mx'}} - \underbrace{k\ln \lambda}_{\textrm{'c'}}
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| \end{align}
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| </math>
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| which can be seen to be in the standard form of a straight line. Therefore if the data came from a Weibull distribution then a straight line is expected on a Weibull plot. | |
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| There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using <math>\hat F = \frac{i-0.3}{n+0.4}</math> where <math>i</math> is the rank of the data point and <math>n</math> is the number of data points.<ref>Wayne Nelson (2004) ''Applied Life Data Analysis''. Wiley-Blackwell ISBN 0-471-64462-5</ref>
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| Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter <math>k</math> and the scale parameter <math>\lambda</math> can also be inferred.
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| in survival
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| The Weibull distribution is used{{Citation needed|date=June 2010}} | |
| * In [[survival analysis]]<ref>[http://www.statsoft.com/textbook/survival-failure-time-analysis/#distribution Survival/Failure Time Analysis]</ref>
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| * In [[reliability engineering]] and [[failure analysis]]
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| * In [[industrial engineering]] to represent [[manufacturing]] and [[Delivery (commerce)|delivery]] times
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| * In [[extreme value theory]]
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| * In [[weather forecasting]]
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| ** To describe [[Wind power#Distribution of wind speed|wind speed distributions]], as the natural distribution often matches the Weibull shape<ref>[http://www.reuk.co.uk/Wind-Speed-Distribution-Weibull.htm Wind Speed Distribution Weibull]</ref>
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| * In communications systems engineering
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| ** In [[radar]] systems to model the dispersion of the received signals level produced by some types of clutters
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| ** 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
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| [[File:FitWeibullDistr.tif|thumb|240px|Fitted cumulative Weibull distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]
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| * In [[general insurance]] to model the size of [[reinsurance]] claims, and the cumulative development of [[asbestosis]] losses
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| * In forecasting technological change (also known as the Sharif-Islam model)<ref>{{cite web|url=http://www.sciencedirect.com/science/article/pii/0040162580900268 |title=The Weibull distribution as a general model for forecasting technological change |publisher=Sciencedirect.com |date= |accessdate=2013-09-05}}</ref>
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| * In [[hydrology]] the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Weibull distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
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| *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 Rosin-Rammler distribution.{{Citation needed|date = May 2011}} In this context it predicts fewer fine particles than the [[Log-normal distribution]] and it is generally most accurate for narrow particle size distributions.{{Citation needed|date = May 2011}} The interpretation of the cumulative distribution function is that ''F''(''x; k; λ'') is the [[mass fraction]] of particles with diameter smaller than ''x'', where ''λ'' is the mean particle size and ''k'' is a measure of the spread of particle sizes.
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| ==Related distributions==
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| *The translated Weibull distribution contains an additional parameter.<ref name="JKB"/> It has the [[probability density function]]
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| :<math>f(x;k,\lambda, \theta)={k \over \lambda} \left({x - \theta \over \lambda}\right)^{k-1} e^{-({x-\theta \over \lambda})^k}\,</math>
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| for <math>x \geq \theta</math> and ''f''(''x''; ''k'', λ, θ) = 0 for ''x'' < θ, where <math>k >0</math> is the [[shape parameter]], <math>\lambda >0</math> is the [[scale parameter]] and <math>\theta</math> is the [[location parameter]] of the distribution. When θ=0, this reduces to the 2-parameter distribution.
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| *The Weibull distribution can be characterized as the distribution of a random variable ''X'' such that the random variable
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| :<math>Y = \left(\frac{X}{\lambda}\right)^k</math>
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| is the standard [[exponential distribution]] with intensity 1.<ref name="JKB"/>
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| *The Weibull distribution interpolates between the exponential distribution with intensity 1/λ when ''k'' = 1 and a [[Rayleigh distribution]] of mode <math>\sigma = \lambda/\sqrt{2}</math> when ''k'' = 2.
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| *The Weibull distribution can also be characterized in terms of a [[Uniform distribution (continuous)|uniform distribution]]: if ''X'' is uniformly distributed on (0,1), then the random variable <math>\lambda(-\ln(1-X))^{1/k}\,</math> is Weibull distributed with parameters ''k'' and λ.{{Citation needed|date=April 2012}} This leads to an easily implemented numerical scheme for simulating a Weibull distribution.
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| *The Weibull distribution (usually sufficient in [[reliability engineering]]) is a special case of the three parameter [[exponentiated Weibull distribution]] where the additional exponent equals 1. The exponentiated Weibull distribution accommodates [[Unimodal function|unimodal]], [[Bathtub curve|bathtub shaped]]*<ref>{{cite web|url=http://www.sys-ev.com/reliability01.htm|title=System evolution and reliability of systems|publisher=Sysev (Belgium)|date=2010-01-01}}</ref> and [[Monotonic function|monotone]] failure [[failure rate|rate]]s.
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| *The Weibull distribution is a special case of the [[generalized extreme value distribution]]. It was in this connection that the distribution was first identified by [[Maurice Fréchet]] in 1927.<ref>{{cite book|last=Montgomery|first=Douglas|title=Introduction to statistical quality control|publisher=John Wiley|location=[S.l.]|isbn=9781118146811|page=95}}</ref> The closely related [[Fréchet distribution]], named for this work, has the probability density function
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| :<math>f_{\rm{Frechet}}(x;k,\lambda)=\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{-1-k} e^{-(x/\lambda)^{-k}} = *f_{\rm{Weibull}}(x;-k,\lambda).</math>
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| *The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is a [[poly-Weibull distribution]].
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| *The Weibull distribution was first applied by {{harvtxt|Rosin|Rammler|1933}} to describe particle size distributions. It is widely used in [[mineral processing]] to describe [[particle size distribution]]s in [[comminution]] processes. In this context the cumulative distribution is given by
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| :<math>f(x;P_{\rm{80}},m) = \begin{cases}
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| 1-e^{ln\left(0.2\right)\left(\frac{x}{P_{\rm{80}}}\right)^m} & x\geq0 ,\\
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| 0 & x<0 ,\end{cases}</math>
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| where
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| :<math>x</math>: Particle size
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| :<math>P_{\rm{80}}</math>: 80<sup>th</sup> percentile of the particle size distribution
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| :<math>m</math>: Parameter describing the spread of the distribution
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| ==See also==
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| * [[Fisher–Tippett–Gnedenko theorem]]
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| * [[Logistic distribution]]
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| * [[particle-size distribution#Rosin-Rammler distribution|Rosin-Rammler distribution]] for particle size analysis
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| ==References==
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| {{Reflist}}
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| ==Bibliography==
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| *{{citation|last=Fréchet|first=Maurice|authorlink=Maurice Fréchet|title=Sur la loi de probabilité de l'écart maximum|journal=Annales de la Société Polonaise de Mathematique, Cracovie|volume=6|pages=93–116|year=1927}}.
| |
| *{{Citation | last1=Johnson | first1=Norman L. | last2=Kotz | first2=Samuel | last3=Balakrishnan | first3=N. | title=Continuous univariate distributions. Vol. 1 | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | series=Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics | isbn=978-0-471-58495-7 | mr = 1299979| year=1994}}
| |
| *{{Citation | first1=G.|last1 = Muraleedharan | first2=A.G.|last2=Rao|first3=P.G.|last3=Kurup|first4=N. Unnikrishnan|last4=Nair|first5=Mourani|last5=Sinha| year = 2007 | title = Coastal Engineering | issue=8 | volume=54 |pages = 630–638|doi=10.1016/j.coastaleng.2007.05.001 | journal=Coastal Engineering}}
| |
| *{{citation|last1=Rosin|first1=P.|last2=Rammler|first2=E.|year=1933|title=The Laws Governing the Fineness of Powdered Coal|journal=Journal of the Institute of Fuel|volume=7|pages=29–36}}.
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| *{{Citation | last1=Sagias | first1=Nikos C. | last2=Karagiannidis | first2=George K. | title=Gaussian class multivariate Weibull distributions: theory and applications in fading channels | doi=10.1109/TIT.2005.855598 | id={{MathSciNet | id = 2237527}} | year=2005 | journal=Institute of Electrical and Electronics Engineers. Transactions on Information Theory | issn=0018-9448 | volume=51 | issue=10 | pages=3608–3619 | url = http://pelopas.uop.gr/~nsagias/Files/Papers/Journals/2005/J4_2005.pdf}}
| |
| *{{citation|last=Weibull|first=W.|authorlink=Waloddi Weibull|year=1951|title=A statistical distribution function of wide applicability|journal=J. Appl. Mech.-Trans. ASME|volume=18|issue=3|pages=293–297|url=http://www.barringer1.com/wa_files/Weibull-ASME-Paper-1951.pdf}}.
| |
| *{{cite web|url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm|chapter=Weibull Distribution|title=Engineering statistics handbook|publisher=[[National Institute of Standards and Technology]]|year=2008}}
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| *{{cite web |url=http://www.erpt.org/014Q/nelsa-06.htm |title=Dispersing Powders in Liquids, Part 1, Chap 6: Particle Volume Distribution |accessdate=2008-02-05 |last=Nelson, Jr |first=Ralph |date=2008-02-05}}
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| ==External links==
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| * {{springer|title=Weibull distribution|id=p/w097370}}
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| * [http://www.mathpages.com/home/kmath122/kmath122.htm Mathpages - Weibull Analysis]
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| * [http://reliawiki.org/index.php/The_Weibull_Distribution The Weibull Distribution]
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| * [http://www.crgraph.com/Weibull11e.pdf Reliability Analysis with Weibull]
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| * Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Weibull Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Survival analysis]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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