#### 52.2.15 Rayleigh Random Variable

The Rayleigh distribution coincides with the \(\chi^2\) distribution with two degrees of freedom.

Function: pdf_rayleigh (x,b)

  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)

  Categories:Package distrib ·

Function: cdf_rayleigh (x,b)

  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)

  Categories:Package distrib ·

Function: quantile_rayleigh (q,b)

  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

  Categories:Package distrib ·

Function: mean_rayleigh (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

  Categories:Package distrib ·

Function: var_rayleigh (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

  Categories:Package distrib ·

Function: std_rayleigh (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

  Categories:Package distrib ·

Function: skewness_rayleigh (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

  Categories:Package distrib ·

Function: kurtosis_rayleigh (b)

  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

  Categories:Package distrib ·

Function: random_rayleigh (b)
`random_rayleigh` (b,n)

  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")`.

  Categories:Package distrib ·Random numbers ·