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} |