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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").
Categories: Package distrib
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.
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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
Categories: Package distrib
Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib
Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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)
Categories: Package distrib
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)
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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.
Categories: Package distrib
Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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.
Categories: Package distrib
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
Categories: Package distrib
Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib ·Random numbers
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").
Categories: Package distrib
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").
Categories: Package distrib
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").
Categories: Package distrib
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.
Categories: Package distrib
Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib
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.
Categories: Package distrib
Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Categories: Package distrib ·Package distrib
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").
Categories: Package distrib ·Random numbers
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