Functions and Variables for continuous distributions (Maxima Manual)

41.2 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").

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) assume(s>0)$ cdf_normal(x,m,s);
                             x - m
                       erf(---------)
                           sqrt(2) s    1
(%o3)                  -------------- + -
                             2          2

See also erf.

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

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").

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").

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").

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").

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").

Function: random_normal (m, s) ¶

Function: 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").

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").

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

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").

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").

Function: var_student_t (n) ¶

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

(%i1) load ("distrib")$
(%i2) assume(n>2)$  var_student_t(n);
                                n
(%o3)                         -----
                              n - 2

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").

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").

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").

Function: random_student_t (n) ¶

Function: 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").

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));

       .01370030107589574 sqrt(5)
(%o2)  --------------------------
       sqrt(2) sqrt(14) sqrt(%pi)
   1.654562884111515E-4 sqrt(5)
 + ----------------------------
            sqrt(%pi)
   .02434921505438663 sqrt(5)
 + --------------------------
              %pi

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

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 if the global variable numer equals true or at least one of the arguments is a float, otherwise it returns a nominal expression.

(%i1) load ("distrib")$
(%i2) cdf_noncentral_student_t(-2,5,-5);
(%o2) cdf_noncentral_student_t(- 2, 5, - 5)
(%i3) cdf_noncentral_student_t(-2.0,5,-5);
(%o3)          .9952030093319743

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").

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) (assume(df>1), mean_noncentral_student_t(df,k));
                   df - 1
             gamma(------) sqrt(df) k
                     2
(%o2)        ------------------------
                              df
                sqrt(2) gamma(--)
                              2

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").

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").

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").

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").

Function: random_noncentral_student_t (n, ncp) ¶

Function: 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").

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma density is returned.

(%i1) load ("distrib")$
(%i2) pdf_chi2(x,n);
                                    n
(%o2)                  pdf_gamma(x, -, 2)
                                    2
(%i3) assume(x>0, n>0)$  pdf_chi2(x,n);
                         n/2 - 1   - x/2
                        x        %e
(%o4)                   ----------------
                          n/2       n
                         2    gamma(-)
                                    2

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

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 if the global variable numer equals true, otherwise it returns a nominal expression based on the gamma quantile function, since the \(Chi^2(n)\) random variable is equivalent to the \(Gamma(n/2,2)\).

(%i1) load ("distrib")$
(%i2) quantile_chi2(0.99,9);
(%o2)                   21.66599433346194
(%i3) quantile_chi2(0.99,n);
                                        n
(%o3)              quantile_gamma(0.99, -, 2)
                                        2

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma mean is returned.

(%i1) load ("distrib")$
(%i2) mean_chi2(n);
                                   n
(%o2)                   mean_gamma(-, 2)
                                   2
(%i3) assume(n>0)$ mean_chi2(n);
(%o4)                           n

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma variance is returned.

(%i1) load ("distrib")$
(%i2) var_chi2(n);
                                   n
(%o2)                    var_gamma(-, 2)
                                   2
(%i3) assume(n>0)$ var_chi2(n);
(%o4)                          2 n

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma standard deviation is returned.

(%i1) load ("distrib")$
(%i2) std_chi2(n);
                                   n
(%o2)                    std_gamma(-, 2)
                                   2
(%i3) assume(n>0)$ std_chi2(n);
(%o4)                    sqrt(2) sqrt(n)

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma skewness coefficient is returned.

(%i1) load ("distrib")$
(%i2) skewness_chi2(n);
                                     n
(%o2)                 skewness_gamma(-, 2)
                                     2
(%i3) assume(n>0)$ skewness_chi2(n);
                            2 sqrt(2)
(%o4)                       ---------
                             sqrt(n)

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 \(Gamma(n/2,2)\), therefore when Maxima has not enough information to get the result, a noun form based on the gamma kurtosis coefficient is returned.

(%i1) load ("distrib")$
(%i2) kurtosis_chi2(n);
                                     n
(%o2)                 kurtosis_gamma(-, 2)
                                     2
(%i3) assume(n>0)$ kurtosis_chi2(n);

                               12
(%o4)                          --
                               n

Function: random_chi2 (n) ¶

Function: 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").

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").

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").

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 if the global variable numer equals true, otherwise it returns a nominal expression.

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 \(ncp>=0\).

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 \(ncp>=0\).

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 \(ncp>=0\).

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 \(ncp>=0\).

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 \(ncp>=0\).

Function: random_noncentral_chi2 (n, ncp) ¶

Function: 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").

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").

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

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 \([0,1]\).

This function has no closed form and it is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.

(%i1) load ("distrib")$
(%i2) quantile_f(2/5,sqrt(3),5);
                               2
(%o2)               quantile_f(-, sqrt(3), 5)
                               5
(%i3) %,numer;
(%o3)                   0.518947838573693

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").

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").

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").

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").

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").

Function: random_f (m, n) ¶

Function: 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").

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 \(Weibull(1,1/m)\), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.

(%i1) load ("distrib")$
(%i2) pdf_exp(x,m);
                                        1
(%o2)                 pdf_weibull(x, 1, -)
                                        m
(%i3) assume(x>0,m>0)$  pdf_exp(x,m);
                                - m x
(%o4)                       m %e

Function: cdf_exp (x, m) ¶

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

(%i1) load ("distrib")$
(%i2) cdf_exp(x,m);
                                        1
(%o2)                 cdf_weibull(x, 1, -)
                                        m
(%i3) assume(x>0,m>0)$  cdf_exp(x,m);
                                 - m x
(%o4)                      1 - %e

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 \([0,1]\).

(%i1) load ("distrib")$
(%i2) quantile_exp(0.56,5);
(%o2)                   .1641961104139661
(%i3) quantile_exp(0.56,m);
                                            1
(%o3)             quantile_weibull(0.56, 1, -)
                                            m

Function: mean_exp (m) ¶

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

(%i1) load ("distrib")$
(%i2) mean_exp(m);
                                       1
(%o2)                  mean_weibull(1, -)
                                       m
(%i3) assume(m>0)$  mean_exp(m);
                                1
(%o4)                           -
                                m

Function: var_exp (m) ¶

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

(%i1) load ("distrib")$
(%i2) var_exp(m);
                                       1
(%o2)                   var_weibull(1, -)
                                       m
(%i3) assume(m>0)$  var_exp(m);
                               1
(%o4)                          --
                                2
                               m

Function: std_exp (m) ¶

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

(%i1) load ("distrib")$
(%i2) std_exp(m);
                                       1
(%o2)                   std_weibull(1, -)
                                       m
(%i3) assume(m>0)$  std_exp(m);
                                1
(%o4)                           -
                                m

Function: skewness_exp (m) ¶

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

(%i1) load ("distrib")$
(%i2) skewness_exp(m);
                                         1
(%o2)                skewness_weibull(1, -)
                                         m
(%i3) assume(m>0)$  skewness_exp(m);
(%o4)                           2

Function: kurtosis_exp (m) ¶

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

(%i1) load ("distrib")$
(%i2) kurtosis_exp(m);
                                         1
(%o2)                kurtosis_weibull(1, -)
                                         m
(%i3) assume(m>0)$  kurtosis_exp(m);
(%o4)                           6

Function: random_exp (m) ¶

Function: 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").

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").

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) assume(x>0, s>0)$  cdf_lognormal(x,m,s);
                           log(x) - m
                       erf(----------)
                           sqrt(2) s     1
(%o3)                  --------------- + -
                              2          2

See also erf.

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

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").

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").

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").

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").

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").

Function: random_lognormal (m, s) ¶

Function: 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").

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").

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

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").

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").

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").

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").

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").

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").

Function: random_gamma (a, b) ¶

Function: 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").

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").

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

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").

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").

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").

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").

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").

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").

Function: random_beta (a, b) ¶

Function: 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").

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").

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").

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").

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").

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").

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").

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").

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").

Function: random_continuous_uniform (a, b) ¶

Function: 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").

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").

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").

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").

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").

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").

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").

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").

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").

Function: random_logistic (a, b) ¶

Function: 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").

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").

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").

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").

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").

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").

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").

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").

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").

Function: random_pareto (a, b) ¶

Function: 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").

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").

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").

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").

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").

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").

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").

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").

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").

Function: random_weibull (a, b) ¶

Function: 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").

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 \(Weibull(2,1/b)\), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.

(%i1) load ("distrib")$
(%i2) pdf_rayleigh(x,b);
                                        1
(%o2)                 pdf_weibull(x, 2, -)
                                        b
(%i3) assume(x>0,b>0)$ pdf_rayleigh(x,b);
                                    2  2
                           2     - b  x
(%o4)                   2 b  x %e

Function: cdf_rayleigh (x, b) ¶

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

(%i1) load ("distrib")$
(%i2) cdf_rayleigh(x,b);
                                        1
(%o2)                 cdf_weibull(x, 2, -)
                                        b
(%i3) assume(x>0,b>0)$ cdf_rayleigh(x,b);
                                   2  2
                                - b  x
(%o4)                     1 - %e

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 \([0,1]\).

(%i1) load ("distrib")$
(%i2) quantile_rayleigh(0.99,b);
                                            1
(%o2)             quantile_weibull(0.99, 2, -)
                                            b
(%i3) assume(x>0,b>0)$ quantile_rayleigh(0.99,b);
                        2.145966026289347
(%o4)                   -----------------
                                b

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 \(Weibull(2,1/b)\), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull mean is returned.

(%i1) load ("distrib")$
(%i2) mean_rayleigh(b);
                                       1
(%o2)                  mean_weibull(2, -)
                                       b
(%i3) assume(b>0)$ mean_rayleigh(b);
                            sqrt(%pi)
(%o4)                       ---------
                               2 b

Function: var_rayleigh (b) ¶

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

(%i1) load ("distrib")$
(%i2) var_rayleigh(b);
                                       1
(%o2)                   var_weibull(2, -)
                                       b
(%i3) assume(b>0)$ var_rayleigh(b);

                                 %pi
                             1 - ---
                                  4
(%o4)                        -------
                                2
                               b

Function: std_rayleigh (b) ¶

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

(%i1) load ("distrib")$
(%i2) std_rayleigh(b);
                                       1
(%o2)                   std_weibull(2, -)
                                       b
(%i3) assume(b>0)$ std_rayleigh(b);
                                   %pi
                          sqrt(1 - ---)
                                    4
(%o4)                     -------------
                                b

Function: skewness_rayleigh (b) ¶

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

(%i1) load ("distrib")$
(%i2) skewness_rayleigh(b);
                                         1
(%o2)                skewness_weibull(2, -)
                                         b
(%i3) assume(b>0)$ skewness_rayleigh(b);
                         3/2
                      %pi      3 sqrt(%pi)
                      ------ - -----------
                        4           4
(%o4)                 --------------------
                               %pi 3/2
                          (1 - ---)
                                4

Function: kurtosis_rayleigh (b) ¶

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

(%i1) load ("distrib")$
(%i2) kurtosis_rayleigh(b);
                                         1
(%o2)                kurtosis_weibull(2, -)
                                         b
(%i3) assume(b>0)$ kurtosis_rayleigh(b);
                                  2
                             3 %pi
                         2 - ------
                               16
(%o4)                    ---------- - 3
                              %pi 2
                         (1 - ---)
                               4

Function: random_rayleigh (b) ¶

Function: 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").

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").

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").

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").

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").

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").

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").

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").

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").

Function: random_laplace (a, b) ¶

Function: 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").

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").

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").

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").

Function: random_cauchy (a, b) ¶

Function: 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").

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").

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").

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").

Function: mean_gumbel (a, b) ¶

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

(%i1) load ("distrib")$
(%i2) assume(b>0)$  mean_gumbel(a,b);
(%o3)                     %gamma b + a

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

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").

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").

Function: skewness_gumbel (a, b) ¶

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

(%i1) load ("distrib")$
(%i2) assume(b>0)$ skewness_gumbel(a,b);
                       12 sqrt(6) zeta(3)
(%o3)                  ------------------
                                 3
                              %pi
(%i4) numer:true$ skewness_gumbel(a,b);
(%o5)                   1.139547099404649

where zeta stands for the Riemann’s zeta function.

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").

Function: random_gumbel (a, b) ¶

Function: 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").