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The library does not check for all possible errors in the

input parameters of the methods.

For the cases that a check is done and the parameter does not fit –

the method returns the value -999.

In several cases the algorithm for the inverse is trying

the forward method and the result may include a residual error.

If accuracy is important re-check the result by implementing

the forward method.

Some of the algorithms are taken from this site :

algorithms

See here Beta_function

of the number of successes in a sequence of n independent yes/no experiments,

each of which yields success with probability p.

Such a success/failure experiment is also called a Bernoulli experiment

or Bernoulli trial.

The binomial distribution is the basis for the popular binomial test

of statistical significance.

It is frequently used to model number of successes in a sample

of size n from a population of size N.

Since the samples are not independent (this is sampling without replacement),

the resulting distribution is a hypergeometric distribution,

not a binomial one. However, for N much larger than n,

the binomial distribution is a good approximation, and widely used.

For further information see

Binomial_distribution

The method returns an array of two Double variables, the first

is the Binomial probability for k out of n trials

with probability of p for each trial, and the second –

the Binomial Mass function for the number k.

with k degrees of freedom is the distribution of a sum of the squares

of k independent standard normal random variables.

It is one of the most widely used probability distributions

in inferential statistics, e.g. in hypothesis testing,

or in construction of confidence intervals.

The best-known situations in which the Chi-square distribution

is used are the common Chi-square tests for goodness of fit

of an observed distribution to a theoretical one, and of the

independence of two criteria of classification of qualitative data.

Many other statistical tests also lead to a use of this distribution,

like Friedman's analysis of variance by ranks.

For further information see

Chi square distribution

This method returns an array of two Double variables,

the first is the probability that the sum of n squares of

normal distributed variables (n Degrees Of Freedom) will be less than

x, and the second is the density function for x.

with wide applicability primarily due to its relation to the exponential and Gamma distributions.

The Erlang distribution was developed by A. K. Erlang to examine

the number of telephone calls which might be made at the same time

to the operators of the switching stations.

This work on telephone traffic engineering has been expanded to consider

waiting times in queueing systems in general.

The distribution is now used in the fields of stochastic processes

and of biomathematics.

The distribution is a continuous distribution, which has a positive value

for all real numbers greater than zero, and is given by two parameters:

the shape k, which is a non-negative integer,

and the rate λ, which is a non-negative real number.

The distribution is sometimes defined using the inverse of the rate parameter,

the scale μ.

For further information see

Erlang distribution

The method returns an array of two Double variables,

the first is the Erlang probability for the number x

and the second is the Density F(x).

are a class of continuous probability distributions.

They describe the times between events in a Poisson process,

i.e. a process in which events occur continuously and independently

at a constant average rate.

For further information look here:

Exponential distribution

The method returns an array of two Double variables, the first is

the Exponential probability for the number x with the parameter

Lambda and the second is the Density function.

It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution

(after R.A. Fisher and George W. Snedecor).

The F-distribution arises frequently as the null distribution

of a test statistic, especially in likelihood-ratio tests,

perhaps most notably in the analysis of variance.

It provides the probability for the ratio of two Chi-squared variables,

each with its DOF.

The probability is for variables larger than the defined X

(mode "R").

To have the left part use 1-Pf in the application.

For further information see

F distribution

Returns an array of two Double variables,

the first is the Fisher F probability for n1 and n2 Degrees Of Freedom

for the number f and the second is the Density function.

See here Gamma_function

distributions:

1. The probability distribution of the number X of Bernoulli trials

needed to get one success, supported on the set { 1, 2, 3, ...}

2. The probability distribution of the number Y = X − 1 of

failures before the first success, supported on the set { 0, 1, 2, 3, ... }

Which of these one calls "the" geometric distribution is a matter of convention

and convenience.

These two different geometric distributions should not be confused with each other.

Often, the name shifted geometric distribution is adopted for the first one

(distribution of the number X); however, to avoid ambiguity,

it is considered wise to indicate which is intended, by mentioning the range explicitly.

The library use the parameter Shifted as explained above,

with values 0 for not shifted (var. Y) and 1 for shifted, var X .

For further information see

Geometric_distribution

The method returns an array of two Double variables, the first

is the Geometric probability for k trials with probability of p

for each trial, and the second the Geometric Mass function for the number k.

with n trials and trial’s probability p, is Pb.

(n Degrees Of Freedom) of which probability is Pc.

with trial’s probability p, is Pg.

Lambda is Pp.

according to the DOF and the selected Mode.

returns the number of different possibilities to choose

a group of k out of a group of n.

The formula is n! / (k! * (n-k)! ).

See here N_choose_k

is an absolutely continuous probability distribution.

The graph of the associated probability density function is

"bell"-shaped, with peak at the mean, and is known as the

Gaussian function or bell curve.

The parameters μ and σ² are the mean and the variance.

The distribution with μ = 0 and σ² = 1 is called standard normal.

For further information see

Normal distribution

The library enables four types of results for the distribution,

named Modes:

1. The probability that the random variable is less than z,

defined as Left, "L".

2. The probability that the random variable is larger than z,

defined as Right, "R".

3. The probability that the random variable is between –z and z,

defined as Center, "C".

4. The probability that the random variable is less than –z

or Larger than z, defined as Tails, "L".

The method returns an array of two Double variables,

the first is the Normal Probability P(z) and the second is the Density F(z).

that expresses the probability of a number of events occurring

in a fixed period of time if these events occur with a known average rate

and independently of the time since the last event.

(The Poisson distribution can also be used for the number of events

in other specified intervals such as distance, area or volume.)

For further information see

Poisson distribution

The method returns an array of two Double variables,

the first is the Poisson Probability Pp(k) and

the second is the value of the Poisson Mass function for k.

Lambda is the time interval.

As an example of how it arises, the wind speed will have a Rayleigh distribution

if the components of the two-dimensional wind velocity vector are uncorrelated

and normally distributed with equal variance.

The distribution is named after Lord Rayleigh.

The distribution is a family of distributions, different by the Scale factor.

For further information see

Rayleigh_distribution

The method returns an array of two Double variables,

the first is the Rayleigh probability for the number r and

the second is the Density F(r).

is a continuous probability distribution that arises in

the problem of estimating the mean of a normally distributed

population when the sample size is small.

It is the basis of the popular Student's t-tests for the

statistical significance of the difference between two sample

means, and for confidence intervals for the difference between

two population means.

For further information see

t distribution

The library enable four types of results for the distribution, named Modes:

1. The probability that the random variable is less than z,

defined as Left, "L".

2. The probability that the random variable is larger than z,

defined as Right, "R".

3. The probability that the random variable is between –z and z,

defined as Center, "C".

4. The probability that the random variable is less than –z or

Larger than z, defined as Tails, "T".

The method returns an array of two Number variables,

the first is the Student t probability for n Degrees Of Freedom

for the number x with the selected Mode,

and the second is the density function for x.

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