### 52.2 Functions and Variables for continuous distributions

Function: pdf_normal (x,m,s)

  Returns the value at x of the density function of a Normal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: cdf_normal (x,m,s)

  Returns the value at x of the distribution function of a Normal(m,s) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function `erf`.

      (%i1) load ("distrib")$
      (%i2) cdf_normal(x,m,s);
                                          x - m
                                    erf(---------)
                                        sqrt(2) s    1
      (%o2)                         -------------- + -
                                          2          2

  See also `erf`.

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  Categories: Package distrib

Function: quantile_normal (q,m,s)

  Returns the q-quantile of a Normal(m,s) random variable, with s>0; in other words, this is the inverse of `cdf_normal`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

      (%i1) load ("distrib")$
      (%i2) quantile_normal(95/100,0,1);
                                            9
      (%o2)             sqrt(2) inverse_erf(--)
                                            10
      (%i3) float(%);
      (%o3)               1.644853626951472

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  Categories: Package distrib

Function: mean_normal (m,s)

  Returns the mean of a Normal(m,s) random variable, with s>0, namely m. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: var_normal (m,s)

  Returns the variance of a Normal(m,s) random variable, with s>0, namely s^2. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: std_normal (m,s)

  Returns the standard deviation of a Normal(m,s) random variable, with s>0, namely s. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_normal (m,s)

  Returns the skewness coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_normal (m,s)

  Returns the kurtosis coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: random_normal (m,s)
`random_normal` (m,s,n)

  Returns a Normal(m,s) random variate, with s>0. Calling `random_normal` with a third argument n, a random sample of size n will be simulated.

  This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_student_t (x,n)

  Returns the value at x of the density function of a Student random variable t(n), with n>0 degrees of freedom. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: cdf_student_t (x,n)

  Returns the value at x of the distribution function of a Student random variable t(n), with n>0 degrees of freedom.

      (%i1) load ("distrib")$
      (%i2) cdf_student_t(1/2, 7/3);
                                               7  1  28
                   beta_incomplete_regularized(-, -, --)
                                               6  2  31
      (%o2)    1 - -------------------------------------
                                     2
      (%i3) float(%);
      (%o3)                .6698450596140415

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  Categories: Package distrib

Function: quantile_student_t (q,n)

  Returns the q-quantile of a Student random variable t(n), with n>0; in other words, this is the inverse of `cdf_student_t`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: mean_student_t (n)

  Returns the mean of a Student random variable t(n), with n>0, which is always equal to 0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: var_student_t (n)

  Returns the variance of a Student random variable t(n), with n>2.

      (%i1) load ("distrib")$
      (%i2) var_student_t(n);
                                      n
      (%o2)                         -----
                                    n - 2

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  Categories: Package distrib

Function: std_student_t (n)

  Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_student_t (n)

  Returns the skewness coefficient of a Student random variable t(n), with n>3, which is always equal to 0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_student_t (n)

  Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: random_student_t (n)
`random_student_t` (n,m)

  Returns a Student random variate t(n), with n>0. Calling `random_student_t` with a second argument m, a random sample of size m will be simulated.

  The implemented algorithm is based on the fact that if Z is a normal random variable N(0,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                                 Z
                       X = -------------
                           /   2  \ 1/2
                           |  S   |
                           | ---  |
                           \  n   /

  is a Student random variable with n degrees of freedom, t(n).

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

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  Categories: Package distrib ·Random numbers

Function: pdf_noncentral_student_t (x,n,ncp)

  Returns the value at x of the density function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

  Sometimes an extra work is necessary to get the final result.

      (%i1) load ("distrib")$
      (%i2) expand(pdf_noncentral_student_t(3,5,0.1));

                                 7/2                         7/2
            0.04296414417400905 5      1.323650307289301e-6 5
      (%o2) ------------------------ + -------------------------
               3/2   5/2                       sqrt(%pi)
              2    14    sqrt(%pi)
                                                              7/2
                                         1.94793720435093e-4 5
                                       + ------------------------
                                                   %pi

      (%i3) float(%);
      (%o3)          .02080593159405669

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  Categories: Package distrib

Function: cdf_noncentral_student_t (x,n,ncp)

  Returns the value at x of the distribution function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. This function has no closed form and it is numerically computed.

      (%i1) load ("distrib")$
      (%i2) cdf_noncentral_student_t(-2,5,-5);
      (%o2)          .9952030093319743

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  Categories: Package distrib

Function: quantile_noncentral_student_t (q,n,ncp)

  Returns the q-quantile of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp; in other words, this is the inverse of `cdf_noncentral_student_t`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: mean_noncentral_student_t (n,ncp)

  Returns the mean of a noncentral Student random variable nc_t(n,ncp), with n>1 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

      (%i1) load ("distrib")$
      (%i2) mean_noncentral_student_t(df,k);
                         df - 1
                   gamma(------) sqrt(df) k
                           2
      (%o2)        ------------------------
                                    df
                      sqrt(2) gamma(--)
                                    2

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  Categories: Package distrib

Function: var_noncentral_student_t (n,ncp)

  Returns the variance of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: std_noncentral_student_t (n,ncp)

  Returns the standard deviation of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_noncentral_student_t (n,ncp)

  Returns the skewness coefficient of a noncentral Student random variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_noncentral_student_t (n,ncp)

  Returns the kurtosis coefficient of a noncentral Student random variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: random_noncentral_student_t (n,ncp)
`random_noncentral_student_t` (n,ncp,m)

  Returns a noncentral Student random variate nc_t(n,ncp), with n>0. Calling `random_noncentral_student_t` with a third argument m, a random sample of size m will be simulated.

  The implemented algorithm is based on the fact that if X is a normal random variable N(ncp,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                                 X
                       U = -------------
                           /   2  \ 1/2
                           |  S   |
                           | ---  |
                           \  n   /

  is a noncentral Student random variable with n degrees of freedom and noncentrality parameter ncp, nc_t(n,ncp).

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

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  Categories: Package distrib ·Random numbers

Function: pdf_chi2 (x,n)

  Returns the value at x of the density function of a Chi-square random variable Chi^2(n), with n>0. The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) pdf_chi2(x,n);
                               n/2 - 1   - x/2
                              x        %e
      (%o2)                   ----------------
                                n/2       n
                               2    gamma(-)
                                          2

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  Categories: Package distrib

Function: cdf_chi2 (x,n)

  Returns the value at x of the distribution function of a Chi-square random variable Chi^2(n), with n>0.

      (%i1) load ("distrib")$
      (%i2) cdf_chi2(3,4);
                                                     3
      (%o2)      1 - gamma_incomplete_regularized(2, -)
                                                     2
      (%i3) float(%);
      (%o3)               .4421745996289256

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  Categories: Package distrib

Function: quantile_chi2 (q,n)

  Returns the q-quantile of a Chi-square random variable Chi^2(n), with n>0; in other words, this is the inverse of `cdf_chi2`. Argument q must be an element of [0,1].

  This function has no closed form and it is numerically computed.

      (%i1) load ("distrib")$
      (%i2) quantile_chi2(0.99,9);
      (%o2)                   21.66599433346194

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  Categories: Package distrib

Function: mean_chi2 (n)

  Returns the mean of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) mean_chi2(n);
      (%o2)                           n

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  Categories: Package distrib

Function: var_chi2 (n)

  Returns the variance of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) var_chi2(n);
      (%o2)                          2 n

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  Categories: Package distrib

Function: std_chi2 (n)

  Returns the standard deviation of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) std_chi2(n);
      (%o2)                    sqrt(2) sqrt(n)

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  Categories: Package distrib

Function: skewness_chi2 (n)

  Returns the skewness coefficient of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) skewness_chi2(n);
                                           3/2
                                          2
      (%o2)                              -------
                                         sqrt(n)

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  Categories: Package distrib

Function: kurtosis_chi2 (n)

  Returns the kurtosis coefficient of a Chi-square random variable Chi^2(n), with n>0.

  The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

      (%i1) load ("distrib")$
      (%i2) kurtosis_chi2(n);
                                     12
      (%o2)                          --
                                     n

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  Categories: Package distrib

Function: random_chi2 (n)
`random_chi2` (n,m)

  Returns a Chi-square random variate Chi^2(n), with n>0. Calling `random_chi2` with a second argument m, a random sample of size m will be simulated.

  The simulation is based on the Ahrens-Cheng algorithm. See `random_gamma` for details.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the density function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: cdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the distribution function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: quantile_noncentral_chi2 (q,n,ncp)

  Returns the q-quantile of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in other words, this is the inverse of `cdf_noncentral_chi2`. Argument q must be an element of [0,1].

  This function has no closed form and it is numerically computed.

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  Categories: Package distrib

Function: mean_noncentral_chi2 (n,ncp)

  Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

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  Categories: Package distrib

Function: var_noncentral_chi2 (n,ncp)

  Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

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  Categories: Package distrib

Function: std_noncentral_chi2 (n,ncp)

  Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

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  Categories: Package distrib

Function: skewness_noncentral_chi2 (n,ncp)

  Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

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  Categories: Package distrib

Function: kurtosis_noncentral_chi2 (n,ncp)

  Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

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  Categories: Package distrib

Function: random_noncentral_chi2 (n,ncp)
`random_noncentral_chi2` (n,ncp,m)

  Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. Calling `random_noncentral_chi2` with a third argument m, a random sample of size m will be simulated.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_f (x,m,n)

  Returns the value at x of the density function of a F random variable F(m,n), with m,n>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: cdf_f (x,m,n)

  Returns the value at x of the distribution function of a F random variable F(m,n), with m,n>0.

      (%i1) load ("distrib")$
      (%i2) cdf_f(2,3,9/4);
                                               9  3  3
      (%o2)    1 - beta_incomplete_regularized(-, -, --)
                                               8  2  11
      (%i3) float(%);
      (%o3)                 0.66756728179008

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  Categories: Package distrib

Function: quantile_f (q,m,n)

  Returns the q-quantile of a F random variable F(m,n), with m,n>0; in other words, this is the inverse of `cdf_f`. Argument q must be an element of [0,1].

      (%i1) load ("distrib")$
      (%i2) quantile_f(2/5,sqrt(3),5);
      (%o2)                   0.518947838573693

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  Categories: Package distrib

Function: mean_f (m,n)

  Returns the mean of a F random variable F(m,n), with m>0, n>2. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: var_f (m,n)

  Returns the variance of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: std_f (m,n)

  Returns the standard deviation of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_f (m,n)

  Returns the skewness coefficient of a F random variable F(m,n), with m>0, n>6. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_f (m,n)

  Returns the kurtosis coefficient of a F random variable F(m,n), with m>0, n>8. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: random_f (m,n)
`random_f` (m,n,k)

  Returns a F random variate F(m,n), with m,n>0. Calling `random_f` with a third argument k, a random sample of size k will be simulated.

  The simulation algorithm is based on the fact that if X is a Chi^2(m) random variable and Y is a Chi^2(n) random variable, then

                              n X
                          F = ---
                              m Y

  is a F random variable with m and n degrees of freedom, F(m,n).

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

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  Categories: Package distrib ·Random numbers

Function: pdf_exp (x,m)

  Returns the value at x of the density function of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) pdf_exp(x,m);
                                      - m x
      (%o2)                       m %e

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  Categories: Package distrib

Function: cdf_exp (x,m)

  Returns the value at x of the distribution function of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) cdf_exp(x,m);
                                       - m x
      (%o2)                      1 - %e

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  Categories: Package distrib

Function: quantile_exp (q,m)

  Returns the q-quantile of an Exponential(m) random variable, with m>0; in other words, this is the inverse of `cdf_exp`. Argument q must be an element of [0,1].

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) quantile_exp(0.56,5);
      (%o2)                   .1641961104139661
      (%i3) quantile_exp(0.56,m);
                                   0.8209805520698303
      (%o3)                        ------------------
                                           m

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  Categories: Package distrib

Function: mean_exp (m)

  Returns the mean of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) mean_exp(m);
                                      1
      (%o2)                           -
                                      m

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  Categories: Package distrib

Function: var_exp (m)

  Returns the variance of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) var_exp(m);
                                     1
      (%o2)                          --
                                      2
                                     m

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  Categories: Package distrib

Function: std_exp (m)

  Returns the standard deviation of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) std_exp(m);
                                      1
      (%o2)                           -
                                      m

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  Categories: Package distrib

Function: skewness_exp (m)

  Returns the skewness coefficient of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) skewness_exp(m);
      (%o2)                           2

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  Categories: Package distrib

Function: kurtosis_exp (m)

  Returns the kurtosis coefficient of an Exponential(m) random variable, with m>0.

  The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

      (%i1) load ("distrib")$
      (%i2) kurtosis_exp(m);
      (%o3)                           6

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  Categories: Package distrib

Function: random_exp (m)
`random_exp` (m,k)

  Returns an Exponential(m) random variate, with m>0. Calling `random_exp` with a second argument k, a random sample of size k will be simulated.

  The simulation algorithm is based on the general inverse method.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_lognormal (x,m,s)

  Returns the value at x of the density function of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: cdf_lognormal (x,m,s)

  Returns the value at x of the distribution function of a Lognormal(m,s) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function `erf`.

      (%i1) load ("distrib")$
      (%i2) cdf_lognormal(x,m,s);

                                 log(x) - m
                             erf(----------)
                                 sqrt(2) s     1
      (%o2)                  --------------- + -
                                    2          2

  See also `erf`.

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  Categories: Package distrib

Function: quantile_lognormal (q,m,s)

  Returns the q-quantile of a Lognormal(m,s) random variable, with s>0; in other words, this is the inverse of `cdf_lognormal`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

      (%i1) load ("distrib")$
      (%i2) quantile_lognormal(95/100,0,1);
                        sqrt(2) inverse_erf(9/10)
      (%o2)           %e
      (%i3) float(%);
      (%o3)               5.180251602233015

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  Categories: Package distrib

Function: mean_lognormal (m,s)

  Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: var_lognormal (m,s)

  Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: std_lognormal (m,s)

  Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_lognormal (m,s)

  Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_lognormal (m,s)

  Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: random_lognormal (m,s)
`random_lognormal` (m,s,n)

  Returns a Lognormal(m,s) random variate, with s>0. Calling `random_lognormal` with a third argument n, a random sample of size n will be simulated.

  Log-normal variates are simulated by means of random normal variates. See `random_normal` for details.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_gamma (x,a,b)

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

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  Categories: Package distrib

Function: cdf_gamma (x,a,b)

  Returns the value at x of the distribution function of a Gamma(a,b) random variable, with a,b>0.

      (%i1) load ("distrib")$
      (%i2) cdf_gamma(3,5,21);
                                                    1
      (%o2)     1 - gamma_incomplete_regularized(5, -)
                                                    7
      (%i3) float(%);
      (%o3)              4.402663157376807E-7

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  Categories: Package distrib

Function: quantile_gamma (q,a,b)

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

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  Categories: Package distrib

Function: mean_gamma (a,b)

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

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  Categories: Package distrib

Function: var_gamma (a,b)

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

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  Categories: Package distrib

Function: std_gamma (a,b)

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

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  Categories: Package distrib

Function: skewness_gamma (a,b)

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

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  Categories: Package distrib

Function: kurtosis_gamma (a,b)

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

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  Categories: Package distrib

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

  Returns a Gamma(a,b) random variate, with a,b>0. Calling `random_gamma` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is a combination of two procedures, depending on the value of parameter a:

  For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.

  For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial cdf_tributions. Computing, 12, 223-246.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_beta (x,a,b)

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

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  Categories: Package distrib

Function: cdf_beta (x,a,b)

  Returns the value at x of the distribution function of a Beta(a,b) random variable, with a,b>0.

      (%i1) load ("distrib")$
      (%i2) cdf_beta(1/3,15,2);
                                   11
      (%o2)                     --------
                                14348907
      (%i3) float(%);
      (%o3)              7.666089131388195E-7

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  Categories: Package distrib

Function: quantile_beta (q,a,b)

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

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  Categories: Package distrib

Function: mean_beta (a,b)

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

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  Categories: Package distrib

Function: var_beta (a,b)

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

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  Categories: Package distrib

Function: std_beta (a,b)

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

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  Categories: Package distrib

Function: skewness_beta (a,b)

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

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  Categories: Package distrib

Function: kurtosis_beta (a,b)

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

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  Categories: Package distrib

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

  Returns a Beta(a,b) random variate, with a,b>0. Calling `random_beta` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322

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

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  Categories: Package distrib ·Random numbers

Function: pdf_continuous_uniform (x,a,b)

  Returns the value at x of the density function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: cdf_continuous_uniform (x,a,b)

  Returns the value at x of the distribution function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: quantile_continuous_uniform (q,a,b)

  Returns the q-quantile of a Continuous Uniform(a,b) random variable, with a<b; in other words, this is the inverse of `cdf_continuous_uniform`. 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_continuous_uniform (a,b)

  Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: var_continuous_uniform (a,b)

  Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: std_continuous_uniform (a,b)

  Returns the standard deviation of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: skewness_continuous_uniform (a,b)

  Returns the skewness coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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Function: kurtosis_continuous_uniform (a,b)

  Returns the kurtosis coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first `load("distrib")`.

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

  Returns a Continuous Uniform(a,b) random variate, with a<b. Calling `random_continuous_uniform` with a third argument n, a random sample of size n will be simulated.

  This is a direct application of the `random` built-in Maxima function.

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

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  Categories: Package distrib ·Random numbers

Function: pdf_logistic (x,a,b)

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

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Function: cdf_logistic (x,a,b)

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

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Function: quantile_logistic (q,a,b)

  Returns the q-quantile of a Logistic(a,b) random variable , with b>0; in other words, this is the inverse of `cdf_logistic`. 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_logistic (a,b)

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

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Function: var_logistic (a,b)

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

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Function: std_logistic (a,b)

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

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Function: skewness_logistic (a,b)

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

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Function: kurtosis_logistic (a,b)

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

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

  Returns a Logistic(a,b) random variate, with b>0. Calling `random_logistic` 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_pareto (x,a,b)

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

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Function: cdf_pareto (x,a,b)

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

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Function: quantile_pareto (q,a,b)

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

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  Categories: Package distrib

Function: mean_pareto (a,b)

  Returns the mean of a Pareto(a,b) random variable, with a>1,b>0. To make use of this function, write first `load("distrib")`.

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Function: var_pareto (a,b)

  Returns the variance of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: std_pareto (a,b)

  Returns the standard deviation of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: skewness_pareto (a,b)

  Returns the skewness coefficient of a Pareto(a,b) random variable, with a>3,b>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

Function: kurtosis_pareto (a,b)

  Returns the kurtosis coefficient of a Pareto(a,b) random variable, with a>4,b>0. To make use of this function, write first `load("distrib")`.

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  Categories: Package distrib

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

  Returns a Pareto(a,b) random variate, with a>0,b>0. Calling `random_pareto` 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_weibull (x,a,b)

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

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  Categories: Package distrib

Function: cdf_weibull (x,a,b)

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

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Function: quantile_weibull (q,a,b)

  Returns the q-quantile of a 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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  Categories: Package distrib

Function: mean_weibull (a,b)

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

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  Categories: Package distrib

Function: var_weibull (a,b)

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

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  Categories: Package distrib

Function: std_weibull (a,b)

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

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Function: skewness_weibull (a,b)

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

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Function: kurtosis_weibull (a,b)

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

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

  Returns a 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_rayleigh (x,b)

  Returns the value at x of the density function of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) pdf_rayleigh(x,b);
                                          2  2
                                 2     - b  x
      (%o2)                   2 b  x %e

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  Categories: Package distrib

Function: cdf_rayleigh (x,b)

  Returns the value at x of the distribution function of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) cdf_rayleigh(x,b);
                                         2  2
                                      - b  x
      (%o2)                     1 - %e

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Function: quantile_rayleigh (q,b)

  Returns the q-quantile of a 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 Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) quantile_rayleigh(0.99,b);
                              2.145966026289347
      (%o2)                   -----------------
                                      b

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  Categories: Package distrib

Function: mean_rayleigh (b)

  Returns the mean of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) mean_rayleigh(b);
                                  sqrt(%pi)
      (%o2)                       ---------
                                     2 b

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  Categories: Package distrib

Function: var_rayleigh (b)

  Returns the variance of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) var_rayleigh(b);
                                       %pi
                                   1 - ---
                                        4
      (%o2)                        -------
                                      2
                                     b

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  Categories: Package distrib

Function: std_rayleigh (b)

  Returns the standard deviation of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) std_rayleigh(b);
                                         %pi
                                sqrt(1 - ---)
                                          4
      (%o2)                     -------------
                                      b

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  Categories: Package distrib

Function: skewness_rayleigh (b)

  Returns the skewness coefficient of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) skewness_rayleigh(b);
                               3/2
                            %pi      3 sqrt(%pi)
                            ------ - -----------
                              4           4
      (%o2)                 --------------------
                                     %pi 3/2
                                (1 - ---)
                                      4

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  Categories: Package distrib

Function: kurtosis_rayleigh (b)

  Returns the kurtosis coefficient of a Rayleigh(b) random variable, with b>0.

  The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

      (%i1) load ("distrib")$
      (%i2) kurtosis_rayleigh(b);
                                        2
                                   3 %pi
                               2 - ------
                                     16
      (%o2)                    ---------- - 3
                                    %pi 2
                               (1 - ---)
                                     4

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  Categories: Package distrib

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

  Returns a 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_laplace (x,a,b)

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

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Function: cdf_laplace (x,a,b)

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

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  Categories: Package distrib

Function: quantile_laplace (q,a,b)

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

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  Categories: Package distrib

Function: mean_laplace (a,b)

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

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Function: var_laplace (a,b)

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

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Function: std_laplace (a,b)

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

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Function: skewness_laplace (a,b)

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

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Function: kurtosis_laplace (a,b)

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

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

  Returns a Laplace(a,b) random variate, with b>0. Calling `random_laplace` 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_cauchy (x,a,b)

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

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Function: cdf_cauchy (x,a,b)

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

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  Categories: Package distrib

Function: quantile_cauchy (q,a,b)

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

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  Categories: Package distrib

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

  Returns a Cauchy(a,b) random variate, with b>0. Calling `random_cauchy` 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")`.

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  Categories: Package distrib ·Random numbers

Function: pdf_gumbel (x,a,b)

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

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Function: cdf_gumbel (x,a,b)

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

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Function: quantile_gumbel (q,a,b)

  Returns the q-quantile of a Gumbel(a,b) random variable, with b>0; in other words, this is the inverse of `cdf_gumbel`. 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_gumbel (a,b)

  Returns the mean of a Gumbel(a,b) random variable, with b>0.

      (%i1) load ("distrib")$
      (%i2) mean_gumbel(a,b);
      (%o2)                     %gamma b + a

  where symbol `%gamma` stands for the Euler-Mascheroni constant. See also `%gamma`.

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Function: var_gumbel (a,b)

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

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Function: std_gumbel (a,b)

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

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Function: skewness_gumbel (a,b)

  Returns the skewness coefficient of a Gumbel(a,b) random variable, with b>0.

      (%i1) load ("distrib")$
      (%i2) skewness_gumbel(a,b);
                                        3/2
                                     2 6    zeta(3)
      (%o2)                          --------------
                                             3
                                          %pi

  where `zeta` stands for the Riemann’s zeta function.

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Function: kurtosis_gumbel (a,b)

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

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  Categories: Package distrib ·Package distrib

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

  Returns a Gumbel(a,b) random variate, with b>0. Calling `random_gumbel` 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")`.

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  Categories: Package distrib ·Random numbers