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47. distrib


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47.1 Introduction to distrib

Package distrib contains a set of functions for making probability computations on both discrete and continuous univariate models.

What follows is a short reminder of basic probabilistic related definitions.

Let f(x) be the density function of an absolute continuous random variable X. The distribution function is defined as

                       x
                      /
                      [
               F(x) = I     f(u) du
                      ]
                      /
                       minf

which equals the probability Pr(X <= x).

The mean value is a localization parameter and is defined as

                     inf
                    /
                    [
           E[X]  =  I   x f(x) dx
                    ]
                    /
                     minf

The variance is a measure of variation,

                 inf
                /
                [                    2
         V[X] = I     f(x) (x - E[X])  dx
                ]
                /
                 minf

which is a positive real number. The square root of the variance is the standard deviation, D[X]=sqrt(V[X]), and it is another measure of variation.

The skewness coefficient is a measure of non-symmetry,

                 inf
                /
            1   [                    3
  SK[X] = ----- I     f(x) (x - E[X])  dx
              3 ]
          D[X]  /
                 minf

And the kurtosis coefficient measures the peakedness of the distribution,

                 inf
                /
            1   [                    4
  KU[X] = ----- I     f(x) (x - E[X])  dx - 3
              4 ]
          D[X]  /
                 minf

If X is gaussian, KU[X]=0. In fact, both skewness and kurtosis are shape parameters used to measure the non-gaussianity of a distribution.

If the random variable X is discrete, the density, or probability, function f(x) takes positive values within certain countable set of numbers x_i, and zero elsewhere. In this case, the distribution function is

                       ====
                       \
                F(x) =  >    f(x )
                       /        i
                       ====
                      x <= x
                       i

The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form

                       ====
                       \
                E[X] =  >  x  f(x ) ,
                       /    i    i
                       ====
                        x 
                         i
                ====
                \                     2
        V[X] =   >    f(x ) (x - E[X])  ,
                /        i    i
                ====
                 x
                  i
               D[X] = sqrt(V[X]),
                     ====
              1      \                     3
  SK[X] =  -------    >    f(x ) (x - E[X])  
           D[X]^3    /        i    i
                     ====
                      x
                       i

and

                     ====
              1      \                     4
  KU[X] =  -------    >    f(x ) (x - E[X])   - 3 ,
           D[X]^4    /        i    i
                     ====
                      x
                       i

respectively.

Package distrib includes functions for simulating random variates. Some of these functions make use of optional variables indicating the algorithm to be used. The general inverse method (based on the fact that if u is an uniform random number in (0,1), then F^(-1)(u) is a random variate with distribution F) is implemented in most cases; this is a suboptimal method in terms of timing, but useful for comparing with other algorithms. In this example, the performance of algorithms ahrens_cheng and inverse for simulating chi-square variates are compared by means of their histograms:

(%i1) load(distrib)$
(%i2) load(descriptive)$
(%i3) showtime: true$
Evaluation took 0.00 seconds (0.00 elapsed) using 32 bytes.
(%i4) random_chi2_algorithm: 'ahrens_cheng$
                                       histogram(random_chi2(10,500))$
Evaluation took 0.00 seconds (0.00 elapsed) using 40 bytes.
Evaluation took 0.69 seconds (0.71 elapsed) using 5.694 MB.
(%i6) random_chi2_algorithm: 'inverse$ histogram(random_chi2(10,500))$
Evaluation took 0.00 seconds (0.00 elapsed) using 32 bytes.
Evaluation took 10.15 seconds (10.17 elapsed) using 322.098 MB.

In order to make visual comparisons among algorithms for a discrete variate, function barsplot of the descriptive package should be used.

Note that some work remains to be done, since these simulating functions are not yet checked by more rigurous goodness of fit tests.

Please, consult an introductory manual on probability and statistics for more information about all this mathematical stuff.

There is a naming convention in package distrib. Every function name has two parts, the first one makes reference to the function or parameter we want to calculate,

Functions:
   Density function            (pdf_*)
   Distribution function       (cdf_*)
   Quantile                    (quantile_*)
   Mean                        (mean_*)
   Variance                    (var_*)
   Standard deviation          (std_*)
   Skewness coefficient        (skewness_*)
   Kurtosis coefficient        (kurtosis_*)
   Random variate              (random_*)

The second part is an explicit reference to the probabilistic model,

Continuous distributions:
   Normal              (*normal)
   Student             (*student_t)
   Chi^2               (*chi2)
   F                   (*f)
   Exponential         (*exp)
   Lognormal           (*lognormal)
   Gamma               (*gamma)
   Beta                (*beta)
   Continuous uniform  (*continuous_uniform)
   Logistic            (*logistic)
   Pareto              (*pareto)
   Weibull             (*weibull)
   Rayleigh            (*rayleigh)
   Laplace             (*laplace)
   Cauchy              (*cauchy)
   Gumbel              (*gumbel)

Discrete distributions:
   Binomial             (*binomial)
   Poisson              (*poisson)
   Bernoulli            (*bernoulli)
   Geometric            (*geometric)
   Discrete uniform     (*discrete_uniform)
   hypergeometric       (*hypergeometric)
   Negative binomial    (*negative_binomial)

For example, pdf_student_t(x,n) is the density function of the Student distribution with n degrees of freedom, std_pareto(a,b) is the standard deviation of the Pareto distribution with parameters a and b and kurtosis_poisson(m) is the kurtosis coefficient of the Poisson distribution with mean m.

In order to make use of package distrib you need first to load it by typing

(%i1) load(distrib)$

For comments, bugs or suggestions, please contact the author at 'mario AT edu DOT xunta DOT es'.

Categories:  Statistical functions · Share packages · Package distrib


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47.2 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).

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

See also erf.

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).

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).

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).

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).

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).

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).

Categories:  Package distrib

Option variable: random_normal_algorithm

Default value: box_mueller

This is the selected algorithm for simulating random normal variates. Implemented algorithms are box_mueller and inverse:

See also random_normal.

Categories:  Package 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.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_normal_algorithm, which defaults to box_mueller.

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

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

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. 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) cdf_student_t(1/2, 7/3);
                                     1  7
(%o2)                  cdf_student_t(-, -)
                                     2  3
(%i3) %,numer;
(%o3)                   .6698450596140417

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).

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).

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) assume(n>2)$  var_student_t(n);
                                n
(%o3)                         -----
                              n - 2

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).

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).

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).

Categories:  Package distrib

Option variable: random_student_t_algorithm

Default value: ratio

This is the selected algorithm for simulating random Student variates. Implemented algorithms are inverse and ratio:

See also random_student_t.

Categories:  Package 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.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_student_t_algorithm, which defaults to ratio.

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

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), 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

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.

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

(%i1) load (distrib)$
(%i2) cdf_chi2(3,4);
(%o2)                  cdf_gamma(3, 2, 2)
(%i3) cdf_chi2(3,4),numer;
(%o3)                   .4421745996289249

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 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

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), 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

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), 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

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), 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)

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), 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)

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), 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

Categories:  Package distrib

Option variable: random_chi2_algorithm

Default value: ahrens_cheng

This is the selected algorithm for simulating random Chi-square variates. Implemented algorithms are ahrens_cheng and inverse:

See also random_chi2.

Categories:  Package distrib

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.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_chi2_algorithm, which defaults to ahrens_cheng.

See also random_chi2_algorithm. To make use of this function, write first load(distrib).

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).

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. 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) cdf_f(2,3,9/4);
                                     9
(%o2)                    cdf_f(2, 3, -)
                                     4
(%i3) %,numer;
(%o3)                   0.66756728179008

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].

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

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).

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).

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).

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).

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).

Categories:  Package distrib

Option variable: random_f_algorithm

Default value: inverse

This is the selected algorithm for simulating random F variates. Implemented algorithms are ratio and inverse:

See also random_f.

Categories:  Package 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.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_f_algorithm, which defaults to inverse.

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

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), 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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.

(%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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.

(%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

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), 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_exp(m);
                                       1
(%o2)                  mean_weibull(1, -)
                                       m
(%i3) assume(m>0)$  mean_exp(m);
                                1
(%o4)                           -
                                m

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.

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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.

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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.

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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.

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

Categories:  Package distrib

Option variable: random_exp_algorithm

Default value: inverse

This is the selected algorithm for simulating random exponential variates. Implemented algorithms are inverse, ahrens_cheng and ahrens_dieter

See also random_exp.

Categories:  Package distrib

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.

There are three algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_exp_algorithm, which defaults to inverse.

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

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).

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

See also erf.

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).

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).

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).

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).

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).

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).

Categories:  Package 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. There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_normal_algorithm, which defaults to box_mueller.

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

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).

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.

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) cdf_gamma(3,5,21);
(%o2)                  cdf_gamma(3, 5, 21)
(%i3) %,numer;
(%o3)                 4.402663157135039E-7

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).

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).

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).

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).

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).

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).

Categories:  Package distrib

Option variable: random_gamma_algorithm

Default value: ahrens_cheng

This is the selected algorithm for simulating random gamma variates. Implemented algorithms are ahrens_cheng and inverse

See also random_gamma.

Categories:  Package 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.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_gamma_algorithm, which defaults to ahrens_cheng.

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

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).

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.

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) cdf_beta(1/3,15,2);
                                 1
(%o2)                   cdf_beta(-, 15, 2)
                                 3
(%i3) %,numer;
(%o3)                 7.666089131388224E-7

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).

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).

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).

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).

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).

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).

Categories:  Package distrib

Option variable: random_beta_algorithm

Default value: cheng

This is the selected algorithm for simulating random beta variates. Implemented algorithms are cheng, inverse and ratio

See also random_beta.

Categories:  Package 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.

There are three algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_beta_algorithm, which defaults to cheng.

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

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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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).

Categories:  Package 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).

Categories:  Package 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).

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).

Categories:  Package 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).

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).

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).

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).

Categories:  Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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).

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).

Categories:  Package 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).

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).

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).

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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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), 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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.

(%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

Categories:  Package distrib

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.

(%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

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), 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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.

(%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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.

(%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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.

(%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

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), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.

(%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

Categories:  Package distrib

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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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).

Categories:  Package 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).

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).

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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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).

Categories:  Package 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).

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).

Categories:  Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

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).

Categories:  Package 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).

Categories:  Package 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).

Categories:  Package 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.

Categories:  Package distrib

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).

Categories:  Package 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).

Categories:  Package 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.

Categories:  Package distrib

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).

Categories:  Package distrib · Package 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.

Only the inverse method is implemented. To make use of this function, write first load(distrib).

Categories:  Package distrib · Random numbers


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47.3 Functions and Variables for discrete distributions

Function: pdf_binomial (x,n,p)

Returns the value at x of the probability function of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_binomial (x,n,p)

Returns the value at x of the distribution function of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer.

This function is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.

(%i1) load (distrib)$
(%i2) cdf_binomial(5,7,1/6);
                                         1
(%o2)                 cdf_binomial(5, 7, -)
                                         6
(%i3) cdf_binomial(5,7,1/6), numer;
(%o3)                   .9998713991769548

Categories:  Package distrib

Function: quantile_binomial (q,n,p)

Returns the q-quantile of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer; in other words, this is the inverse of cdf_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_binomial (n,p)

Returns the mean of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: var_binomial (n,p)

Returns the variance of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: std_binomial (n,p)

Returns the standard deviation of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: skewness_binomial (n,p)

Returns the skewness coefficient of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: kurtosis_binomial (n,p)

Returns the kurtosis coefficient of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Option variable: random_binomial_algorithm

Default value: kachit

This is the selected algorithm for simulating random binomial variates. Implemented algorithms are kachit, bernoulli and inverse:

See also random_binomial.

Categories:  Package distrib

Function: random_binomial (n,p)
Function: random_binomial (n,p,m)

Returns a Binomial(n,p) random variate, with 0<p<1 and n a positive integer. Calling random_binomial with a third argument m, a random sample of size m will be simulated.

There are three algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_binomial_algorithm, which defaults to kachit.

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

Categories:  Package distrib · Random numbers

Function: pdf_poisson (x,m)

Returns the value at x of the probability function of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_poisson (x,m)

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

This function is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.

(%i1) load (distrib)$
(%i2) cdf_poisson(3,5);
(%o2)                   cdf_poisson(3, 5)
(%i3) cdf_poisson(3,5), numer;
(%o3)                   .2650259152973617

Categories:  Package distrib

Function: quantile_poisson (q,m)

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

Categories:  Package distrib

Function: mean_poisson (m)

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

Categories:  Package distrib

Function: var_poisson (m)

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

Categories:  Package distrib

Function: std_poisson (m)

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

Categories:  Package distrib

Function: skewness_poisson (m)

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

Categories:  Package distrib

Function: kurtosis_poisson (m)

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

Categories:  Package distrib

Option variable: random_poisson_algorithm

Default value: ahrens_dieter

This is the selected algorithm for simulating random Poisson variates. Implemented algorithms are ahrens_dieter and inverse:

See also random_poisson.

Categories:  Package distrib

Function: random_poisson (m)
Function: random_poisson (m,n)

Returns a Poisson(m) random variate, with m>0. Calling random_poisson with a second argument n, a random sample of size n will be simulated.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_poisson_algorithm, which defaults to ahrens_dieter.

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

Categories:  Package distrib · Random numbers

Function: pdf_bernoulli (x,p)

Returns the value at x of the probability function of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial probability function is returned.

(%i1) load (distrib)$
(%i2) pdf_bernoulli(1,p);
(%o2)                 pdf_binomial(1, 1, p)
(%i3) assume(0<p,p<1)$ pdf_bernoulli(1,p);
(%o4)                           p

Categories:  Package distrib

Function: cdf_bernoulli (x,p)

Returns the value at x of the distribution function of a Bernoulli(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: quantile_bernoulli (q,p)

Returns the q-quantile of a Bernoulli(p) random variable, with 0<p<1; in other words, this is the inverse of cdf_bernoulli. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_bernoulli (p)

Returns the mean of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial mean is returned.

(%i1) load (distrib)$
(%i2) mean_bernoulli(p);
(%o2)                  mean_binomial(1, p)
(%i3) assume(0<p,p<1)$ mean_bernoulli(p);
(%o4)                           p

Categories:  Package distrib

Function: var_bernoulli (p)

Returns the variance of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial variance is returned.

(%i1) load (distrib)$
(%i2) var_bernoulli(p);
(%o2)                  var_binomial(1, p)
(%i3) assume(0<p,p<1)$ var_bernoulli(p);
(%o4)                       (1 - p) p

Categories:  Package distrib

Function: std_bernoulli (p)

Returns the standard deviation of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial standard deviation is returned.

(%i1) load (distrib)$
(%i2) std_bernoulli(p);
(%o2)                  std_binomial(1, p)
(%i3) assume(0<p,p<1)$ std_bernoulli(p);
(%o4)                  sqrt(1 - p) sqrt(p)

Categories:  Package distrib

Function: skewness_bernoulli (p)

Returns the skewness coefficient of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial skewness coefficient is returned.

(%i1) load (distrib)$
(%i2) skewness_bernoulli(p);
(%o2)                skewness_binomial(1, p)
(%i3) assume(0<p,p<1)$ skewness_bernoulli(p);
                             1 - 2 p
(%o4)                  -------------------
                       sqrt(1 - p) sqrt(p)

Categories:  Package distrib

Function: kurtosis_bernoulli (p)

Returns the kurtosis coefficient of a Bernoulli(p) random variable, with 0<p<1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial kurtosis coefficient is returned.

(%i1) load (distrib)$
(%i2) kurtosis_bernoulli(p);
(%o2)                kurtosis_binomial(1, p)
(%i3) assume(0<p,p<1)$ kurtosis_bernoulli(p);
                         1 - 6 (1 - p) p
(%o4)                    ---------------
                            (1 - p) p

Categories:  Package distrib

Function: random_bernoulli (p)
Function: random_bernoulli (p,n)

Returns a Bernoulli(p) random variate, with 0<p<1. Calling random_bernoulli with a second 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

Function: pdf_geometric (x,p)

Returns the value at x of the probability function of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_geometric (x,p)

Returns the value at x of the distribution function of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: quantile_geometric (q,p)

Returns the q-quantile of a Geometric(p) random variable, with 0<p<1; in other words, this is the inverse of cdf_geometric. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_geometric (p)

Returns the mean of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: var_geometric (p)

Returns the variance of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: std_geometric (p)

Returns the standard deviation of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: skewness_geometric (p)

Returns the skewness coefficient of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: kurtosis_geometric (p)

Returns the kurtosis coefficient of a geometric random variable Geo(p), with 0<p<1. To make use of this function, write first load(distrib).

Categories:  Package distrib

Option variable: random_geometric_algorithm

Default value: bernoulli

This is the selected algorithm for simulating random geometric variates. Implemented algorithms are bernoulli, devroye and inverse:

See also random_geometric.

Categories:  Package distrib

Function: random_geometric (p)
Function: random_geometric (p,n)

Returns a Geometric(p) random variate, with 0<p<1. Calling random_geometric with a second argument n, a random sample of size n will be simulated.

There are three algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_geometric_algorithm, which defaults to bernoulli.

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

Categories:  Package distrib · Random numbers

Function: pdf_discrete_uniform (x,n)

Returns the value at x of the probability function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_discrete_uniform (x,n)

Returns the value at x of the distribution function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: quantile_discrete_uniform (q,n)

Returns the q-quantile of a Discrete Uniform(n) random variable, with n a strictly positive integer; in other words, this is the inverse of cdf_discrete_uniform. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_discrete_uniform (n)

Returns the mean of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: var_discrete_uniform (n)

Returns the variance of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: std_discrete_uniform (n)

Returns the standard deviation of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: skewness_discrete_uniform (n)

Returns the skewness coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: kurtosis_discrete_uniform (n)

Returns the kurtosis coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: random_discrete_uniform (n)
Function: random_discrete_uniform (n,m)

Returns a Discrete Uniform(n) random variate, with n a strictly positive integer. Calling random_discrete_uniform with a second argument m, a random sample of size m 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

Function: pdf_hypergeometric (x,n1,n2,n)

Returns the value at x of the probability function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_hypergeometric (x,n1,n2,n)

Returns the value at x of the distribution function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: quantile_hypergeometric (q,n1,n2,n)

Returns the q-quantile of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2; in other words, this is the inverse of cdf_hypergeometric. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_hypergeometric (n1,n2,n)

Returns the mean of a discrete uniform random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: var_hypergeometric (n1,n2,n)

Returns the variance of a hypergeometric random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: std_hypergeometric (n1,n2,n)

Returns the standard deviation of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: skewness_hypergeometric (n1,n2,n)

Returns the skewness coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: kurtosis_hypergeometric (n1,n2,n)

Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).

Categories:  Package distrib

Option variable: random_hypergeometric_algorithm

Default value: kachit

This is the selected algorithm for simulating random hypergeometric variates. Implemented algorithms are kachit and inverse:

See also random_hypergeometric.

Categories:  Package distrib

Function: random_hypergeometric (n1,n2,n)
Function: random_hypergeometric (n1,n2,n,m)

Returns a Hypergeometric(n1,n2,n) random variate, with n1, n2 and n non negative integers and n<=n1+n2. Calling random_hypergeometric with a fourth argument m, a random sample of size m will be simulated.

There are two algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_hypergeometric_algorithm, which defaults to kachit.

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

Categories:  Package distrib · Random numbers

Function: pdf_negative_binomial (x,n,p)

Returns the value at x of the probability function of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: cdf_negative_binomial (x,n,p)

Returns the value at x of the distribution function of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer.

This function is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.

(%i1) load (distrib)$
(%i2) cdf_negative_binomial(3,4,1/8);
                                             1
(%o2)            cdf_negative_binomial(3, 4, -)
                                             8
(%i3) cdf_negative_binomial(3,4,1/8), numer;
(%o3)                  .006238937377929698

Categories:  Package distrib

Function: quantile_negative_binomial (q,n,p)

Returns the q-quantile of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer; in other words, this is the inverse of cdf_negative_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: mean_negative_binomial (n,p)

Returns the mean of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: var_negative_binomial (n,p)

Returns the variance of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: std_negative_binomial (n,p)

Returns the standard deviation of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: skewness_negative_binomial (n,p)

Returns the skewness coefficient of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Function: kurtosis_negative_binomial (n,p)

Returns the kurtosis coefficient of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).

Categories:  Package distrib

Option variable: random_negative_binomial_algorithm

Default value: bernoulli

This is the selected algorithm for simulating random negative binomial variates. Implemented algorithms are devroye, bernoulli and inverse:

See also random_negative_binomial.

Categories:  Package distrib

Function: random_negative_binomial (n,p)
Function: random_negative_binomial (n,p,m)

Returns a Negative Binomial(n,p) random variate, with 0<p<1 and n a positive integer. Calling random_negative_binomial with a third argument m, a random sample of size m will be simulated.

There are three algorithms implemented for this function, the one to be used can be selected giving a certain value to the global variable random_negative_binomial_algorithm, which defaults to bernoulli.

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

Categories:  Package distrib · Random numbers


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