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Encyclopedia > Weibull distribution
 Probability density function Cumulative distribution function Parameters $lambda>0,$ scale (real) $k>0,$ shape (real) Support $x in [0; +infty),$ pdf $(k/lambda) (x/lambda)^{(k-1)} e^{-(x/lambda)^k}$ cdf $1- e^{-(x/lambda)^k}$ Mean $mu=lambda Gammaleft(1+frac{1}{k}right),$ Median $lambdaln(2)^{1/k},$ Mode Variance $sigma^2=lambda^2Gammaleft(1+frac{2}{k}right) - mu^2,$ Skewness $frac{Gamma(1+frac{3}{k})lambda^3-3musigma^2-mu^3}{sigma^3}$ Kurtosis (see text) Entropy $gammaleft(1!-!frac{1}{k}right)+left(frac{lambda}{k}right)^k +lnleft(frac{lambda}{k}right)$ mgf see Weibull fading Char. func.

In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... The Weibull distribution was first introduced by Waloddi Weibull back in 1937 for estimating machinery lifetime and became widely known in 1951. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... Probability theory is the mathematical study of probability. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Ernst Hjalmar Waloddi Weibull (18 June 1887-12 October 1979) was a Swedish engineer, scientist, and mathematician. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

$f(x;k,lambda) = (k/lambda) (x/lambda)^{(k-1)} e^{-(x/lambda)^k},$

where $x geq0$ and k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution.

The cumulative density function is defined as

$F(x;k,lambda) = 1- e^{-(x/lambda)^k},$

where again, x > 0.

The failure rate h (or hazard rate) is given by: Exponential failure density functions A failure rate is the average frequency with which something fails. ...

h(x;k,λ) = (k / λ)(x / λ)(k − 1)

Weibull distributions are often used to model the time until a given technical device fails. If the failure rate of the device decreases over time, one chooses k < 1 (resulting in a decreasing density f). If the failure rate of the device is constant over time, one chooses k = 1, again resulting in a decreasing function f. If the failure rate of the device increases over time, one chooses k > 1 and obtains a density f which increases towards a maximum and then decreases forever. Manufacturers will often supply the shape and scale parameters for the lifetime distribution of a particular device. The Weibull distribution can also be used to model the distribution of wind speeds at a given location on Earth. Again, every location is characterized by a particular shape and scale parameter. Exponential failure density functions A failure rate is the average frequency with which something fails. ... Exponential failure density functions A failure rate is the average frequency with which something fails. ... Exponential failure density functions A failure rate is the average frequency with which something fails. ...

## Contents

The nth raw moment is given by:

$m_n = lambda^n Gamma(1+n/k),$

where Γ is the Gamma function. The expected value and standard deviation of a Weibull random variable can be expressed as: The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability and statistics, the standard deviation is the most common measure of statistical dispersion. ... A random variable is a term used in mathematics and statistics. ...

$textrm{E}(X) = lambda Gamma(1+1/k),$

and

$textrm{var}(X) = lambda^2[Gamma(1+2/k) - Gamma^2(1+1/k)],$

The skewness is given by:

$gamma_1=frac{Gammaleft(1+frac{3}{k}right)lambda^3-3musigma^2-mu^3}{sigma^3}$

The kurtosis excess is given by:

$gamma_2=frac{-6Gamma_1^4+12Gamma_1^2Gamma_2-3Gamma_2^2 -4Gamma_1Gamma_3+Gamma_4}{[Gamma_2-Gamma_1^2]^2}$

where Γi = Γ(1 + i / k). The kurtosis excess may also be written:

$gamma_2=frac{lambda^4Gammaleft(1+frac{4}{k}right) -3sigma^4-4gamma_1sigma^3mu-6sigma^2mu^2-mu^4}{sigma^4}$

## Generating Weibull-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1], then the variate In mathematics, the uniform distributions are simple probability distributions. ...

$X=lambda (-ln(U))^{1/k},$

has a Weibull distribution with parameters k and λ. This follows from the form of the cumulative distribution function.

THIS FUNCTION IS IMPOSSIBLE!!!

## Related distributions

• $X sim mathrm{Exponential}(lambda)$ is an exponential distribution if $X sim mathrm{Weibull}(gamma = 1, lambda^{-1})$.
• $X sim mathrm{Rayleigh}(beta)$ is a Rayleigh distribution if $X sim mathrm{Weibull}(gamma = 2, sqrt{2} beta)$.
• $lambda(-ln(X))^{1/k},$ is a Weibull distribution if $X sim mathrm{Uniform}(0,1)$.

I WOULD LIKE TO SEE JOEL NACHLAS DO THIS!!!! (WHISTLE) In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, FrÃ©chet and Weibull families also known as type I, II and III extreme value distributions. ...

## Uses

Results from FactBites:

 Weibull distribution - Wikipedia, the free encyclopedia (492 words) The Weibull distribution is often used in place of the Normal distribution due to the fact that a Weibull variate can be generated through inversion, while Normal variates are typically generated using the more complicated Box-Muller Method, which requires two uniform random variates. Weibull distributions may also be used to represent manufacturing and delivery times in industrial engineering problems, while it is very important in extreme value theory and weather forecasting. Furthermore, concerning wireless communications, the Weibull distribution may be used for fading channel modelling, since the Weibull fading model seems to exhibit good fit to experimental fading channel measurements.
 Probability distribution - definition of Probability distribution in Encyclopedia (1119 words) A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space. The triangular distribution on [a, b], a special case of which is the distribution of the sum of two uniformly distributed random variables (the convolution of two uniform distributions). The Weibull distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices.
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