Next: Laplace Random Variable, Previous: Weibull Random Variable, Up: Functions and Variables for continuous distributions [Contents][Index]
The Rayleigh distribution coincides with the \(\chi^2\) distribution with two degrees of freedom.
Returns the value at x of the density function of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The pdf is $$ f(x; b) = \cases{ 2b^2 x e^{-b^2 x^2} & for $x \ge 0$ \cr 0 & for $x < 0$ } $$
(%i1) load ("distrib")$
(%i2) pdf_rayleigh(x,b);
                              2  2
                     2     - b  x
(%o2)             2 b  x %e        unit_step(x)
Returns the value at x of the distribution function of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The cdf is $$ F(x; b) = \cases{ 1 - e^{-b^2 x^2} & for $x \ge 0$\cr 0 & for $x < 0$ } $$
(%i1) load ("distrib")$
(%i2) cdf_rayleigh(x,b);
                            2  2
                         - b  x
(%o2)             (1 - %e       ) unit_step(x)
Returns the q-quantile of a 
\({\it Rayleigh}(b)\) random variable, with b>0; in other words, this is the inverse of cdf_rayleigh. Argument q must be an element of [0,1].
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
(%i1) load ("distrib")$
(%i2) quantile_rayleigh(0.99,b);
                        2.145966026289347
(%o2)                   -----------------
                                b
Returns the mean of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The mean is $$ E[X] = {\sqrt{\pi}\over 2b} $$
(%i1) load ("distrib")$
(%i2) mean_rayleigh(b);
                            sqrt(%pi)
(%o2)                       ---------
                               2 b
Returns the variance of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The variance is $$ V[X] = {1\over b^2}\left(1-{\pi \over 4}\right) $$
(%i1) load ("distrib")$
(%i2) var_rayleigh(b);
                                 %pi
                             1 - ---
                                  4
(%o2)                        -------
                                2
                               b
Returns the standard deviation of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The standard deviation is $$ D[X] = {1\over b}\sqrt{\displaystyle 1 - {\pi\over 4}} $$
(%i1) load ("distrib")$
(%i2) std_rayleigh(b);
                                   %pi
                          sqrt(1 - ---)
                                    4
(%o2)                     -------------
                                b
Returns the skewness coefficient of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The skewness coefficient is $$ SK[X] = {2\sqrt{\pi}(\pi - 3)\over (4-\pi)^{3/2}} $$
(%i1) load ("distrib")$
(%i2) skewness_rayleigh(b);
                         3/2
                      %pi      3 sqrt(%pi)
                      ------ - -----------
                        4           4
(%o2)                 --------------------
                               %pi 3/2
                          (1 - ---)
                                4
Returns the kurtosis coefficient of a \({\it Rayleigh}(b)\) random variable, with b>0.
The \({\it Rayleigh}(b)\) random variable is equivalent to the \({\it Weibull}(2,1/b)\).
The kurtosis coefficient is $$ KU[X] = {32-3\pi\over (4-\pi)^2} - 3 $$
(%i1) load ("distrib")$
(%i2) kurtosis_rayleigh(b);
                                  2
                             3 %pi
                         2 - ------
                               16
(%o2)                    ---------- - 3
                              %pi 2
                         (1 - ---)
                               4
Returns a 
\({\it Rayleigh}(b)\) random variate, with b>0. Calling random_rayleigh with a second 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").
Next: Laplace Random Variable, Previous: Weibull Random Variable, Up: Functions and Variables for continuous distributions [Contents][Index]