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

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

Categories: Package distrib ·

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

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

Categories: Package distrib ·

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").

Categories: Package distrib ·

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

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

Categories: Package distrib ·

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

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

Categories: Package distrib ·

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

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

Categories: Package distrib ·

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

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

Categories: Package distrib ·

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

Categories: Package distrib ·

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 ·