Probabilities distribution functions. This unit introduces 35 commonnly used probability distributions.

Discrete and Continuous distributions

A random variable is discrete if it can assume at most a finite or countably infinite number of possible values. A random variable is continuous if it can assume any value in some interval or intervals of real numbers and the probability that it assumes any specific value is zero.

PDF, CDF and inverse CDF

When dealing with the random variable, it is not enough just to determine what values are possible. There is also a need to determine what is probable. We must be able to predict in some sense the values that the variable is likely to assume at any time. Since the behavior of a random variable is governed by chance, these predictions must be made in the face of a great deal of uncertainty. The best that can be done is to describe the behavior of the random variable in terms of probabilities. Two functions are used to accomplish this, probability density functions and cumulative distribution functions.

Discrete distributions: There are several facts to note concerning the PDF in the discrete case. First, the PDF is defined on the entire real line, and for any given real number x, f(x) is the probability that the random variable X assumes the value x. The necessary and sufficient conditions for a function f to be a PDF are:

Even though a discrete density is defined on the entire real line, it is only necessary to specify the density for those values x for which f(x) is not zero.

Continuous distributions: The PDF is defined on an entire real line and is nonnegative. In the continuous case the probability that the random variable X assumes the value x is 0. Therefore, we are interested in finding the probability that random variable X assumes values in some interval [a,b]. The necessary conditions for a function f to be a PDF are:

Note: In Probabilities.pas all probability density functions have a PDF at the end of their name.

Discrete distributions: The word "cumulative" suggests the role of the CDF function. It sums or accumulates the probabilities found by means of PDF. The cumulative distribution function F(x) is defined as follows:

We must sum the PDF over all values of X that occur with nonzero probability that are less than or equal to x.

Continuos distributions: In continuous case is the cumulative distribution function - CDF defined exactly as in the discrete case although found by using integration rather than summation. The cumulative distribution function F(X) is defined as follows:

In this case, to find a specific F(x), we must integrate the density over all real numbers that are less than or equal to x.
Note: In Probabilites.pas all cumulative distribution functions have a CDF at the end of their name.
Generally, the "inverse" cumulative distributions are used to "get" the original values that were used for calculating the CDF. If the distribution is continuous, the result of the inverse CDF is the original value. If the distribution is discrete, the result is generally not the exact original value.
Discrete distributions: As stated above, the relationship between a discrete CDF and its inverse function is more complicated. It is likely that there is no such value x that the CDF of x gives the desired probability. In these cases the inverse function returns the first value x such that the CDF of that value equals or exceeds the desired probability.
Continuos distributions: Calculating the CDF of values gives the probabilities in the interval [0,1]. If we want the original values that were used for the CDF (x), we must apply the inverse cumulative distribution. The result is the original value used to calculate CDF.
Note: In Probabilites.pas all inverse cumulative distribution functions have a CDFInv at the end of their name.

Types

Name	Summary
TDistribution	All supported probability distributions.

Routines

Name	Summary
Beta	Beta function B.
BetaCDF	Beta cumulative distribution function (CDF).
BetaCDFInv	Beta distribution point percent function (PPF).
BetaIncomplete	Incomplete beta function.
BetaPDF	Beta probability density function (PDF).
BetaStat	Beta distribution statistic parameters.
BinomCDF	Binomial cumulative distribution function (CDF).
BinomCDFInv	Binomial distribution point percent function (PPF).
Binomial	Binomial coefficient C(n,k).
BinomPDF	Binomial probability density function (PDF).
BinomStat	Binomial distribution statistic parameters.
BoseEinsteinPDF	Bose-Einstein probability density function (PDF).
CauchyCDF	Cauchy cumulative distribution function (CDF).
CauchyCDFInv	Cauchy distribution point percent function (PPF).
CauchyPDF	Cauchy probability density function (PDF).
CauchyStat	Cauchy distribution statistic parameters.
ChiSquareCDF	Chi-squared cumulative distribution Function (CDF).
ChiSquareCDFInv	Chi-squared distribution point percent function (PPF).
ChiSquarePDF	Chi-squared probability density Function (PDF).
ChiSquareStat	Chi-Squared distribution statistic parameters.
DirichletBeta	Dirichlet Beta function.
DirichletEta	Dirichlet Eta function.
DirichletLambda	Dirichlet Lambda function.
Erf	Error function.
Erfc	The complement to the error function.
ErfInv	Inverse error function.
ErlangCDF	Erlang cumulative distribution function (CDF).
ErlangCDFInv	Erlang distribution point percent function (PPF).
ErlangPDF	Erlang probability density function (PDF).
ErlangStat	Erlang distribution statistic parameters.
ExpCDF	Exponential cumulative distribution function (CDF).
ExpCDFInv	Exponential distribution point percent function (PPF).
ExponentStat	Exponent distribution statistic parameters.
ExpPDF	Exponential probability density function (PDF).
FatigueLifeCDF	Fatigue life cumulative distribution function (CDF).
FatigueLifeCDFInv	Fatigue life distribution point percent function (PPF).
FatigueLifePDF	Fatigue life probability density function (PDF).
FatigueLifeStat	Fatigue life distribution statistic parameters.
FCDF	F cumulative distribution function (CDF).
FCDFInv	F distribution point percent function (PPF).
FermiDiracPDF	Fermi-Dirac probability density function (PDF).
FPDF	F probability density function (PDF).
FStat	F distribution statistic parameters.
GammaCDF	Gamma cumulative distribution function (CDF).
GammaCDFInv	Gamma distribution point percent function (PPF).
GammaFrac	Incomplete Gamma function : continued fraction expansion.
GammaPDF	Gamma probability density function (PDF).
GammaSer	Incomplete Gamma function : series expansion.
GammaStat	Gamma distribution statistic parameters.
GenHarmonic	Generalized harmonic number H(n,m).
GeometricCDF	Geometric cumulative distribution function (CDF).
GeometricCDFInv	Geometric distribution point percent function (PPF).
GeometricPDF	Geometric probability density function (PDF).
GeometricStat	Geometric distribution statistic parameters.
GumbelCDF	Gumbel CDF.
GumbelCDFInv	Gumbel distribution point percent function (PPF).
GumbelPDF	Gumbel PDF.
GumbelStat	Gumbel distribution statistic parameters.
Harmonic	Harmonic number H(n).
HypGeometricCDF	Hypergeometric cumulative distribution function (CDF).
HypGeometricCDFInv	Hypergeometric distribution point percent function (PPF).
HypGeometricPDF	Hypergeometric probability density function (PDF).
HypGeometricStat	Hypergeometric distribution statistic parameters.
IGamma	Incomplete gamma function.
InverseGaussianCDF	Inverse Gaussian cumulative distribution function (CDF).
InverseGaussianCDFInv	Inverse Gaussian distribution point percent function (PPF).
InverseGaussianPDF	Inverse Gaussian probability density function (PDF).
InverseGaussianStat	Inverse Gaussian distribution statistic parameters.
JGamma	Complement to the incomplete gamma function.
JohnsonSBCDF	Johnson bounded cumulative distribution function (CDF).
JohnsonSBCDFInv	Johnson bounded point percent function (PPF).
JohnsonSBPDF	Johnson bounded probability density function (PDF).
JohnsonSBStat	Johnson SB distribution statistic parameters.
JohnsonUBCDF	Johnson unbounded cumulative distribution function (CDF).
JohnsonUBCDFInv	Johnson unbounded point percent function (PPF).
JohnsonUBPDF	Johnson unbounded probability density function (PDF).
JohnsonUBStat	Johnson UB distribution statistic parameters.
LaplaceCDF	Laplace cumulative distribution function (CDF).
LaplaceCDFInv	Laplace distribution point percent function (PPF).
LaplacePDF	Laplace probability densify function (PDF).
LaplaceStat	Laplace distribution statistic parameters.
Lerch	Lerch transcendent function.
LogarithmicCDF	Logarithmic cumulative distribution function (CDF).
LogarithmicCDFInv	Logarithmic point percent function (PPF).
LogarithmicPDF	Logarithmic probability densify function (PDF).
LogarithmicStat	Logarithmic distribution statistic parameters.
LogisticCDF	Logistic cumulative distribution function (CDF).
LogisticCDFInv	Logistic distribution point percent function (PPF).
LogisticPDF	Logistic probability densify function (PDF).
LogisticStat	Logistic distribution statistic parameters.
LogNormalCDF	Log-normal cumulative distribution function (CDF).
LogNormalCDFInv	Log-normal distribution point percent function (PPF).
LogNormalPDF	Log-normal probability densify function (PDF).
LogNormalStat	Log-normal distribution statistic parameters.
LogWeibullCDF	Log-Weibull cumulative distribution function (CDF).
LogWeibullCDFInv	Log-Weibull distribution point percent function (PPF).
LogWeibullPDF	Log-Weibull probability densify function (PDF).
LogWeibullStat	Log-Weibull distribution statistic parameters.
MaxwellCDF	Maxwell cumulative distribution function (CDF).
MaxwellCDFInv	Maxwell distribution point percent function (PPF).
MaxwellPDF	Maxwell probability density function (PDF).
MaxwellStat	Maxwell distribution statistic parameters.
NegBinomCDF	Negative binomial cumulative distribution function (CDF).
NegBinomCDFInv	Negative binomial distribution point percent function (PPF).
NegBinomPDF	Negative binomial probability density function (PDF).
NegBinomStat	Negative binomial distribution statistic parameters.
NormalCDF	Normal cumulative distribution function (CDF).
NormalCDFInv	Normal distribution point percent function (PPF).
NormalPDF	Normal probability density function (PDF).
NormalStat	Normal distribution statistic parameters.
ParetoCDF	Pareto cumulative distribution function (CDF).
ParetoCDFInv	Pareto distribution point percent function (PPF).
ParetoPDF	Pareto probability density function (PDF).
ParetoStat	Pareto distribution statistic parameters.
PoissonCDF	Poisson cumulative distribution function (CDF).
PoissonCDFInv	Poisson distribution point percent function (PPF).
PoissonPDF	Poisson probability density function (PDF).
PoissonStat	Poison distribution statistic parameters.
PowerCDF	Power cumulative density function (CDF).
PowerCDFInv	Power distribution point percent function (PPF).
PowerPDF	Power probability distribution function (PDF).
PowerStat	Power distribution statistic parameters.
RayleighCDF	Rayleigh cumulative distribution function (CDF).
RayleighCDFInv	Rayleigh distribution point percent function (PPF).
RayleighPDF	Rayleigh probability density function (PDF).
RayleighStat	Rayleigh distribution statistic parameters.
RiemannZeta	Riemann Zeta function.
StudentCDF	Student cumulative distribution function (CDF).
StudentCDFInv	Student distribution point percent function (PPF).
StudentPDF	Student probability density function (PDF).
StudentStat	Student distribution statistic parameters.
TriangularCDF	Triangular cumulative distribution function (CDF).
TriangularCDFInv	Triangular distribution point percent function (PPF).
TriangularPDF	Triangular probability density function (PDF).
TriangularStat	Triangular distribution statistic parameters.
UniformCDF	Uniform cumulative distribution function (CDF).
UniformCDFInv	Uniform distribution point percent function (PPF).
UniformDCDF	Discrete uniform cumulative distribution function (CDF).
UniformDCDFInv	Discrete uniform distribution point percent function (PPF).
UniformDPDF	Discrete uniform probability density function (PDF).
UniformDStat	Discrete uniform distribution statistic parameters.
UniformPDF	Uniform probability density function (PDF).
UniformStat	Continuous uniform distribution statistic parameters.
WeibullCDF	Weibull cumulative distribution function (CDF).
WeibullCDFInv	Weibull distribution point percent function (PPF).
WeibullPDF	Weibull probability density function (PDF).
WeibullStat	Weibull distribution statistic parameters.
ZipfCDF	Zipf distribution cumulative distribution function (CDF).
ZipfCDFInv	Zipf distribution point percent function (PPF).
ZipfPDF	Zipf distribution probability density function (PDF).

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