This versatility is one reason for the wide use of the Weibull distribution in reliability. λ are the k σ The mean of \(X\) is \(\displaystyle{\text{E}[X] = \beta\Gamma\left(1+\frac{1}{\alpha}\right)}\). ( The failure rate function \( r \) is given by \[ r(t) = k t^{k-1}, \quad t \in (0, \infty) \]. ( where If \( Z \) has the basic Weibull distribution with shape parameter \( k \) then \( G(Z) \) has the standard uniform distribution. = In the special distribution simulator, select the Weibull distribution. The standard Weibull distribution is the same as the standard exponential distribution. ⁡ If \( U \) has the standard uniform distribution then \( Z = (-\ln U)^{1/k} \) has the basic Weibull distribution with shape parameter \( k \). The first quartile is \( q_1 = (\ln 4 - \ln 3)^{1/k} \). $$f(x) = \left\{\begin{array}{l l} e If \( k = 1 \), \( g \) is decreasing and concave upward with mode \( t = 0 \). It will enhance any encyclopedic page you visit with the magic of the WIKI 2 technology. k Therefore, if the data came from a Weibull distribution then a straight line is expected on a Weibull plot. λ − For selected values of the shape parameter, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation. x For selected values of the parameter, compute the median and the first and third quartiles. Open the special distribution calculator and select the Weibull distribution. \(\E(X) = b \Gamma\left(1 + \frac{1}{k}\right)\), \(\var(X) = b^2 \left[\Gamma\left(1 + \frac{2}{k}\right) - \Gamma^2\left(1 + \frac{1}{k}\right)\right]\), The skewness of \( X \) is \[ \skw(X) = \frac{\Gamma(1 + 3 / k) - 3 \Gamma(1 + 1 / k) \Gamma(1 + 2 / k) + 2 \Gamma^3(1 + 1 / k)}{\left[\Gamma(1 + 2 / k) - \Gamma^2(1 + 1 / k)\right]^{3/2}} \], The kurtosis of \( X \) is \[ \kur(X) = \frac{\Gamma(1 + 4 / k) - 4 \Gamma(1 + 1 / k) \Gamma(1 + 3 / k) + 6 \Gamma^2(1 + 1 / k) \Gamma(1 + 2 / k) - 3 \Gamma^4(1 + 1 / k)}{\left[\Gamma(1 + 2 / k) - \Gamma^2(1 + 1 / k)\right]^2} \]. where It follows that \( U \) has reliability function given by \[ \P(U \gt t) = \left\{\exp\left[-\left(\frac{t}{b}\right)^k\right]\right\}^n = \exp\left[-n \left(\frac{t}{b}\right)^k\right] = \exp\left[-\left(\frac{t}{b / n^{1/k}}\right)^k\right], \quad t \in [0, \infty) \] and so the result follows. Since the Weibull distribution is a scale family for each value of the shape parameter, it is trivially closed under scale transformations. x − ) Except for the point of discontinuity \( t = 1 \), the limits are the CDF of point mass at 1. n Skewness and kurtosis depend only on the standard score of the random variable, and hence are invariant under scale transformations. − {\displaystyle k} parameter given l ) The moments of \(Z\), and hence the mean and variance of \(Z\) can be expressed in terms of the gamma function \( \Gamma \). i θ As k goes to infinity, the Weibull distribution converges to a Dirac delta distribution centered at x = λ. This is because the value of β is equal to the slope of the line in a probability plot. {\displaystyle i} Finally, the Weibull distribution is a member of the family of general exponential distributions if the shape parameter is fixed. W Weibull was not the first person to use the distribution, but was the first to study it extensively and recognize its wide use in applications. Open the special distribution simulator and select the Weibull distribution. x = New content will be added above the current area of focus upon selection {\displaystyle \ln(x)} 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. − The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. The Weibull distribution is used[citation needed], f In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model. λ k , For selected values of the parameters, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.

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