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Parameters λ ∈ ( 0 , + ∞ ) {displaystyle lambda in (0,+infty ),} scale k ∈ ( 0 , + ∞ ) {displaystyle kin (0,+infty ),} shape Support x ∈ [ 0 , + ∞ ) {displaystyle xin [0,+infty ),} PDF f ( x ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k x ≥ 0 0 x < 0 {displaystyle f(x)={egin{cases}{rac {k}{lambda }}left({rac {x}{lambda }}ight)^{k-1}e^{-(x/lambda )^{k}}&xgeq 00&x<0end{cases}}} CDF { 1 − e − ( x / λ ) k x ≥ 0 0 x < 0 {displaystyle {egin{cases}1-e^{-(x/lambda )^{k}}&xgeq 00&x<0end{cases}}} Mean λ Γ ( 1 + 1 / k ) {displaystyle lambda ,Gamma (1+1/k),} Median λ ( ln ( 2 ) ) 1 / k {displaystyle lambda (ln(2))^{1/k},} |
In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.
Contents
- Standard parameterization
- Alternative parameterizations
- Density function
- Cumulative distribution function
- Moments
- Moment generating function
- Shannon entropy
- Maximum likelihood
- Weibull plot
- Related distributions
- References
Standard parameterization
The probability density function of a Weibull random variable is:
where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its 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 and
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:
In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.
Alternative parameterizations
In medical statistics a different parameterization is used. The shape parameter k is the same as above and the scale parameter is
and the probability density function is
The mean is
A third parameterization is sometimes used. In this the shape parameter k is the same as above and the scale parameter is
Density function
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 centered at x = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter.
Cumulative distribution function
The cumulative distribution function for the Weibull distribution is
for x ≥ 0, and F(x; k; λ) = 0 for x < 0.
The quantile (inverse cumulative distribution) function for the Weibull distribution is
for 0 ≤ p < 1.
The failure rate h (or hazard function) is given by
Moments
The moment generating function of the logarithm of a Weibull distributed random variable is given by
where Γ is the gamma function. Similarly, the characteristic function of log X is given by
In particular, the nth raw moment of X is given by
The mean and variance of a Weibull random variable can be expressed as
and
The skewness is given by
where the mean is denoted by μ and the standard deviation is denoted by σ.
The excess kurtosis is given by
where
Moment generating function
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
Alternatively, one can attempt to deal directly with the integral
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. With t replaced by −t, one finds
where G is the Meijer G-function.
The characteristic function has also been obtained by Muraleedharan et al. (2007). The characteristic function and moment generating function of 3-parameter Weibull distribution have also been derived by Muraleedharan & Soares (2014) by a direct approach.
Shannon entropy
The information entropy is given by
where
Maximum likelihood
The maximum likelihood estimator for the
The maximum likelihood estimator for
This being an implicit function, one must generally solve for
When
Also given that condition, the maximum likelihood estimator for
Again, this being an implicit function, one must generally solve for
Weibull plot
The fit of data to a Weibull distribution can be visually assessed using a Weibull Plot. The Weibull Plot is a plot of the empirical cumulative distribution function
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.
There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using
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
The Weibull distribution is used
Related distributions