52.2.14 Weibull Random Variable

Function: pdf_weibull (x,a,b) ¶

Returns the value at x of the density function of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x;a,b)=\{\begin{array}{ll}{\displaystyle \frac{1}{b}{\left(\frac{x}{b}\right)}^{a-1}{e}^{-(x/b{)}^a}}& \text{for }x\ge 0\\ \\ 0& \text{for }x<0\end{array}$$

Categories: Package distrib ·

Function: cdf_weibull (x,a,b) ¶

Returns the value at x of the distribution function of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x;a,b)=\{\begin{array}{ll}1-e^{-(x/b{)}^a}& \text{for }x\ge 0\\ 0& \text{for }x<0\end{array}$$

Categories: Package distrib ·

Function: quantile_weibull (q,a,b) ¶

Returns the q-quantile of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ 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 $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X]=b\mathrm{\Gamma }(1+\frac{1}{a})$$

Categories: Package distrib ·

Function: var_weibull (a,b) ¶

Returns the variance of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X]=b^2[\mathrm{\Gamma }(1+\frac{2}{a})-\mathrm{\Gamma }{(1+\frac{1}{a})}^2]$$

Categories: Package distrib ·

Function: std_weibull (a,b) ¶

Returns the standard deviation of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$D[X]=b\sqrt{\mathrm{\Gamma }(1+\frac{2}{a})-\mathrm{\Gamma }{(1+\frac{1}{a})}^2}$$

Categories: Package distrib ·

Function: skewness_weibull (a,b) ¶

Returns the skewness coefficient of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X]=\frac{{\displaystyle \mathrm{\Gamma }(1+\frac{3}{a})-3\mathrm{\Gamma }(1+\frac{1}{a})\mathrm{\Gamma }(1+\frac{2}{a})+2\mathrm{\Gamma }{(1+\frac{1}{a})}^3}}{{\displaystyle {[\mathrm{\Gamma }(1+\frac{2}{a})-\mathrm{\Gamma }{(1+\frac{1}{a})}^2]}^{3/2}}}$$

Categories: Package distrib ·

Function: kurtosis_weibull (a,b) ¶

Returns the kurtosis coefficient of a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X]=\frac{{\mathrm{\Gamma }}_4-4{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3+6{\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_2-3{\mathrm{\Gamma }}_1^4}{{[{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1^2]}^2}-3$$

where ${\mathrm{\Gamma }}_k=\mathrm{\Gamma }(1+k/a)$ .

Categories: Package distrib ·

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

Returns a $\mathit{W}\mathit{e}\mathit{i}\mathit{b}\mathit{u}\mathit{l}\mathit{l}(a,b)$ 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 ·