#### 52.2.14 Weibull Random Variable

Function: pdf_weibull (x,a,b)

  Returns the value at x of the density function of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The pdf is $$ f(x; a, b) = \cases{ \displaystyle{1\over b} \left({x\over b}\right)^{a-1} e^{-(x/b)^a} & for $x \ge 0$ \cr \cr 0 & for $x < 0$ } $$

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Function: cdf_weibull (x,a,b)

  Returns the value at x of the distribution function of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The cdf is $$ F(x; a, b) = \cases{ 1 - e^{-(x/b)^a} & for $x \ge 0$ \cr 0 & for $x < 0$ } $$

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Function: quantile_weibull (q,a,b)

  Returns the q-quantile of a \({\it Weibull}(a,b)\) random variable, with a,b>0; in other words, this is the inverse of `cdf_weibull`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

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Function: mean_weibull (a,b)

  Returns the mean of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The mean is $$ E[X] = b\Gamma\left(1+{1\over a}\right) $$

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Function: var_weibull (a,b)

  Returns the variance of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The variance is $$ V[X] = b^2\left[\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2\right] $$

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Function: std_weibull (a,b)

  Returns the standard deviation of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The variance is $$ D[X] = b\sqrt{\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2} $$

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Function: skewness_weibull (a,b)

  Returns the skewness coefficient of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The skewness coefficient is $$ SK[X] = {\displaystyle\Gamma\left(1+{3\over a}\right) -3\Gamma\left(1+{1\over a}\right)\Gamma\left(1+{2\over a}\right)+2\Gamma\left(1+{1\over a}\right)^3 \over \displaystyle\left[\Gamma\left(1+{2\over a}\right)-\Gamma\left(1+{1\over a}\right)^2\right]^{3/2} } $$

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Function: kurtosis_weibull (a,b)

  Returns the kurtosis coefficient of a \({\it Weibull}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The kurtosis coefficient is $$ KU[X] = { \Gamma_4 - 4\Gamma_1 \Gamma_3 + 6\Gamma_1^2 \Gamma_2 - 3 \Gamma_1^4 \over \left[\Gamma_2 - \Gamma_1^2\right]^2 } - 3 $$

  where \(\Gamma_k = \Gamma\left(1+k/a\right)\).

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Function: random_weibull (a,b)
`random_weibull` (a,b,n)

  Returns a \({\it Weibull}(a,b)\) random variate, with a,b>0. Calling `random_weibull` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·Random numbers ·