### 52.1 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.

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)
       Noncentral Chi^2    (*noncentral_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)
       Finite discrete      (*general_finite_discrete)

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 ’riotorto AT yahoo DOT com’.

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