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### 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 $$ F\left(x\right)=\int_{ -\infty }^{x}{f\left(u\right)\;du} $$

which equals the probability \({\rm Pr}(X \le x)\) .

The mean value is a localization parameter and is defined as $$ E\left[X\right]=\int_{ -\infty }^{\infty }{x\,f\left(x\right)\;dx} $$

The variance is a measure of variation, $$ V\left[X\right]=\int_{ -\infty }^{\infty }{f\left(x\right)\,\left(x -E\left[X\right]\right)^2\;dx} $$

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, $$ SK\left[X\right]={{\int_{ -\infty }^{\infty }{f\left(x\right)\, \left(x-E\left[X\right]\right)^3\;dx}}\over{D\left[X\right]^3}} $$

And the kurtosis coefficient measures the peakedness of the distribution, $$ KU\left[X\right]={{\int_{ -\infty }^{\infty }{f\left(x\right)\, \left(x-E\left[X\right]\right)^4\;dx}}\over{D\left[X\right]^4}}-3 $$

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 $$ F\left(x\right)=\sum_{x_{i}\leq x}{f\left(x_{i}\right)} $$

The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form $$ \eqalign{ E\left[X\right]&=\sum_{x_{i}}{x_{i}f\left(x_{i}\right)}, \cr V\left[X\right]&=\sum_{x_{i}}{f\left(x_{i}\right)\left(x_{i}-E\left[X\right]\right)^2},\cr D\left[X\right]&=\sqrt{V\left[X\right]},\cr SK\left[X\right]&={{\sum_{x_{i}}{f\left(x\right)\, \left(x-E\left[X\right]\right)^3\;dx}}\over{D\left[X\right]^3}}, \cr KU\left[X\right]&={{\sum_{x_{i}}{f\left(x\right)\, \left(x-E\left[X\right]\right)^4\;dx}}\over{D\left[X\right]^4}}-3, } $$

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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