(maxima.info)Introduction to distrib
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'>.
automatically generated by info2www version 1.2.2.9