Probability density function |
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Cumulative distribution function |
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Parameters | scale (real) shape (real) |
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Support | |
CDF | |
Mean | |
Median | |
Mode | |
Variance | |
Skewness | |
Ex. kurtosis | (see text) |
Entropy | |
MGF | |
CF |
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after 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 the size distribution of particles.
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The probability density function of a Weibull random variable x is:[1]
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).
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.
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.
The cumulative distribution function for the Weibull distribution is
for x ≥ 0, and F(x; k; λ) = 0 for x < 0.
The failure rate h (or hazard rate) is given by
The moment generating function of the logarithm of a Weibull distributed random variable is given by[2]
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 . The kurtosis excess may also be written as:
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.[3] 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 information entropy is given by
where is the Euler–Mascheroni constant.
The goodness of fit of data to a Weibull distribution can be visually assessed using a Weibull Plot.[4] The Weibull Plot is a plot of the empirical cumulative distribution function of data on special axes in a type of Q-Q plot. The axes are versus . The reason for this change of variables is the cumulative distribution function can be linearised:
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 where is the rank of the data point and is the number of data points.[5]
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 and the scale parameter can also be inferred.
The Weibull distribution is used
for and f(x; k, λ, θ) = 0 for x < θ, where is the shape parameter, is the scale parameter and is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.
is the standard exponential distribution with intensity 1.[2]
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