Distributions

Distribution functions are available for an extensive number of probability distributions.  LINGO supports the probability density functions (PDF) for each distribution, as well as their cumulative (CDF) and inverse (INV) functions.   Supported distributions are listed below:

Continuous Distribution

Functions

Description

Parameters and Domain

@PBETACDF( A, B, X)

Cumulative Beta

A = alpha > 0

B = beta > 0

X (0,1)

@PBETAINV( A, B, X)

Inverse Beta

A = alpha > 0

B = beta > 0

X [0,1]

@PBETAPDF( A, B, X)

Beta PDF

A = alpha > 0

B = beta > 0

X (0,1)

@PCACYCDF( L, S, X)

Cumulative Cauchy

L = location

S = scale > 0

X a real

@PCACYINV( L, S, X)

Inverse Cauchy

L = location

S = scale > 0

X [0,1]

@PCACYPDF( L, S, X)

Cauchy PDF

L = location

S = scale > 0

X a real

@PCHISCDF( DF, X)

Cumulative Chi-Square

DF = degrees of freedom = a positive integer

X 0

@PCHISINV( DF, X)

Inverse Chi-Square

DF = degrees of freedom = a positive integer

X [0,1]

@PCHISPDF( DF, X)

Chi-Square PDF

DF = degrees of freedom = a positive integer

X 0

@PEXPOCDF( L, X)

Cumulative Exponential

L = lambda > 0

X 0

@PEXPOINV( L, X)

Inverse Exponential

L = lambda > 0

X [0,1]

@PEXPOPDF( L, X)

Exponential PDF

L = lambda > 0

X 0

@PFDSTCDF( DF1, DF2, X)

Cumulative F-Distribution

DF1,DF2 = degrees of freedom = a positive integer

X 0

@PFDSTINV( DF1, DF2, X)

Inverse F-Distribution

DF1,DF2 = degrees of freedom = a positive integer

X [0,1]

@PFDSTPDF( DF1, DF2, X)

F-Distribution PDF

DF1,DF2 = degrees of freedom = a positive integer

X 0

@PGAMMCDF( SC, SH, X)

Cumulative Gamma

SC = scale > 0

SH = shape > 0

X 0

@PGAMMINV( SC, SH, X)

Inverse Gamma

SC = scale > 0

SH = shape > 0

X [0,1]

@PGAMMPDF( SC, SH, X)

Gamma PDF

SC = scale > 0

SH = shape > 0

X 0

@PGMBLCDF( L, S, X)

Cumulative Gumbel

L = location

S = scale > 0

X a real

@PGMBLINV( L, S, X)

Inverse Gumbel

L = location

S = scale > 0

X [0,1]

@PGMBLPDF( L, S, X)

Gumbel PDF

L = location

S = scale > 0

X a real

@PLAPLCDF( L, S, X)

Cumulative Laplace

L = location

S = scale > 0

X a real

@PLAPLINV( L, S, X)

Inverse Laplace

L = location

S = scale > 0

X [0,1]

@PLAPLPDF( L, S, X)

Laplace PDF

L = location

S = scale > 0

X a real

@PLGSTCDF( L, S, X)

Cumulative Logistic

L = location

S = scale > 0

X a real

@PLGSTINV( L, S, X)

Inverse Logistic

L = location

S = scale > 0

X [0,1]

@PLGSTPDF( L, S, X)

Logistic PDF

L = location

S = scale > 0

X a real

@PLOGNCDF( M, S, X)

Cumulative Lognormal

M = mu

S = sigma > 0

X > 0

@PLOGNINV( M, S, X)

Inverse Lognormal

M = mu

S = sigma > 0

X [0,1]

@PLOGNPDF( M, S, X)

Lognormal PDF

M = mu

S = sigma > 0

X > 0

@PNORMCDF( M, S, X)

Cumulative Normal

M = mu

S = sigma > 0

X a real

@PNORMINV( M, S, X)

Inverse Normal

M = mu

S = sigma > 0

X [0,1]

@PNORMPDF( M, S, X)

Normal PDF

M = mu

S = sigma > 0

X a real

@PPRTOCDF( SC, SH, X)

Cumulative Pareto

SC = scale > 0

SH = shape > 0

X SC

@PPRTOINV( SC, SH, X)

Inverse Pareto

SC = scale > 0

SH = shape > 0

X [0,1]

@PPRTOPDF( SC, SH, X)

Pareto PDF

SC = scale > 0

SH = shape > 0

X SC

@PSMSTCDF( A, X)

Cumulative Symmetric Stable

A = alpha [0.2,2]

X a real

@PSMSTINV( A, X)

Inverse Symmetric Stable

A = alpha [0.2,2]

X [0,1]

@PSMSTPDF( A, X)

Symmetric Stable PDF

A = alpha [0.2,2]

X a real

@PSTUTCDF( DF, X)

Cumulative Student's t

DF = degrees of freedom = a positive integer

X a real

@PSTUTINV( DF, X)

Inverse Student's t

DF = degrees of freedom = a positive integer

X [0,1]

@PSTUTPDF( DF, X)

Student's t PDF

DF = degrees of freedom = a positive integer

X a real

@PTRIACDF( L, U, M, X)

Cumulative Triangular

L = lower limit

U = upper limit

M = mode

X [L,U]

@PTRIAINV( L, U, M, X)

Inverse Triangular

L = lower limit

U = upper limit

M = mode

X [0,1]

@PTRIAPDF( L, U, M, X)

Triangular PDF

L = lower limit

U = upper limit

M = mode

X [L,U]

@PUNIFCDF( L, U, X)

Cumulative Uniform

L = lower limit

U = upper limit

X [L,U]

@PUNIFINV( L, U, X)

Inverse Uniform

L = lower limit

U = upper limit

X [0,1]

@PUNIFPDF( L, U, X)

Uniform PDF

L = lower limit

U = upper limit

X [L,U]

@PWEIBCDF( SC, SH, X)

Cumulative Weibull

SC = scale > 0

SH = shape > 0

X 0

@PWEIBINV( SC, SH, X)

Inverse Weibull

SC = scale > 0

SH = shape > 0

X [0,1]

@PWEIBPDF( SC, SH, X)

Weibull PDF

SC = scale > 0

SH = shape > 0

X 0

 

Discrete Distribution

Functions

Description

Parameters

@PBTBNCDF( N, A, B, X)

Cumulative Beta Binomial

N = trials {0,1,...}

A = alpha (0,+inf)

B = beta (0,+inf)

X {0,1,...,N}

@PBTBNINV( N, A, B, X)

Beta Binomial Inverse

N = trials {0,1,...}

A = alpha (0,+inf)

B = beta (0,+inf)

X [0,1]

@PBTBNPDF( N, A, B, X)

Beta Binomial PDF

N = trials {0,1,...}

A = alpha (0,+inf)

B = beta (0,+inf)

X {0,1,...,N}

@PBINOCDF( N, P, X)

Cumulative Binomial

N = trials {0,1,...}

P = probability of success [0,1]

X {0,1,...,N}

@PBINOINV( N, P, X)

Inverse Binomial

N = trials {0,1,...}

P = probability of success [0,1]

X [0,1]

@PBINOPDF( N, P, X)

Binomial PDF

N = trials  {0,1,...}

P = probability of success [0,1]

X {0,1,...,N}

@PGEOMCDF( P, X)

Cumulative Geometric

P = probability of success (0,1]

X {0,1,...}

@PGEOMINV( P, X)

Inverse Geometric

P = probability of success (0,1]

X [0,1]

@PGEOMPDF( P, X)

Geometric PDF

P = probability of success (0,1]

X {0,1,...}

@PHYPGCDF( N, D, K, X)

Cumulative Hypergeometric

N = population {0,1,...}

D = number defective {0,1,...,N}

K = sample size {0,1,...,N}

X {max(0,D+K-N),...,min(D,K)}

@PHYPGINV( N, D, K, X)

Inverse Hypergeometric

N = population {0,1,...}

D = number defective {0,1,...,N}

K = sample size {0,1,...,N}

X [0,1]

@PHYPGPDF( N, D, K, X)

Hypergeometric PDF

N = population {0,1,...}

D = number defective {0,1,...,N}

K = sample size {0,1,...,N}

X {max(0,D+K-N),...,min(D,K)}

@PLOGRCDF( P, X)

Cumulative Logarithmic

P = p-factor (0,1)

X {0,1,...,N}

@PLOGRINV( P, X)

Inverse Logarithmic

P = p-factor (0,1)

X [0,1]

@PLOGRPDF(  P, X)

Logarithmic PDF

P = p-factor (0,1)

X {0,1,...,N}

@PNEGBCDF( R, P, X)

Cumulative Negative Binomial

R = number of failures (0,+inf)

P = probability of success (0,1)

X {0,1,...}

@PNEGBINV( R, P, X)

Inverse Negative Binomial

R = number of failures (0,+inf)

P = probability of success (0,1)

X [0,1]

@PNEGBPDF( R, P, X)

Negative Binomial PDF

R = number of failures (0,+inf)

P = probability of success (0,1)

X {0,1,...}

@PPOISCDF( L, X)

Cumulative Poisson

L = lambda (0,+inf)

X {0,1,...,N}

@PPOISINV( L, X)

Inverse Poisson

L = lambda (0,+inf)

X [0,1]

@PPOISPDF( L, X)

Poisson PDF

L = lambda (0,+inf)

X {0,1,...,N}