This guide summarizes each continuous distribution currently provided in applpy.distributions.continuous.
Description: Arc-sine distribution on (0, 1) with density concentrated near the boundaries.
Parameters: none.
Typical uses: proportion models with U-shaped behavior and boundary-heavy outcomes.
from applpy.conversion import cdf
from applpy.distributions.continuous import ArcSinRV
x = ArcSinRV()
print(cdf(x, 0.5))Description: Arctangent-based continuous model with tunable location and scale behavior.
Parameters: alpha (scale-like, positive), phi (location/shift).
Typical uses: heavy-tail style modeling when you want a smooth CDF shape.
from applpy.conversion import cdf
from applpy.distributions.continuous import ArcTanRV
x = ArcTanRV(alpha=2, phi=0)
print(cdf(x, 1))Description: Beta distribution on (0, 1) for bounded probabilities and rates.
Parameters: alpha (shape, positive), beta (shape, positive).
Typical uses: Bayesian priors for Bernoulli/binomial rates and bounded random effects.
from applpy.moments import mean
from applpy.distributions.continuous import BetaRV
x = BetaRV(alpha=2, beta=5)
print(mean(x))Description: Joint normal distribution for two correlated continuous variables.
Parameters: mu (shared location), sigma1 (std. dev. of variable 1), sigma2 (std. dev. of variable 2), rho (correlation in [-1, 1]).
Typical uses: correlated measurement errors and two-dimensional Gaussian modeling.
from applpy.distributions.continuous import BivariateNormalRV
x = BivariateNormalRV(mu=0, sigma1=1, sigma2=2, rho=0.5)
print(x)Description: Cauchy distribution with very heavy tails and undefined mean/variance.
Parameters: a (location), alpha (scale, positive).
Typical uses: robust modeling and processes with occasional extreme values.
from applpy.conversion import cdf
from applpy.distributions.continuous import CauchyRV
x = CauchyRV(a=0, alpha=1)
print(cdf(x, 0))Description: Chi distribution for the square root of chi-square variables.
Parameters: N (degrees of freedom, positive integer).
Typical uses: norms of Gaussian vectors and magnitude-type measurements.
from applpy.moments import mean
from applpy.distributions.continuous import ChiRV
x = ChiRV(N=4)
print(mean(x))Description: Chi-square distribution for sums of squared standard normals.
Parameters: N (degrees of freedom, positive integer).
Typical uses: variance inference, goodness-of-fit testing, and likelihood ratio methods.
from applpy.conversion import cdf
from applpy.distributions.continuous import ChiSquareRV
x = ChiSquareRV(N=6)
print(cdf(x, 5))Description: Erlang distribution (Gamma with integer shape) for waiting times.
Parameters: theta (scale, positive), N (shape, positive integer).
Typical uses: queueing and service-time models with staged exponential phases.
from applpy.moments import mean
from applpy.distributions.continuous import ErlangRV
x = ErlangRV(theta=2, N=3)
print(mean(x))Description: Flexible continuous error-family model used in reliability/statistical fitting.
Parameters: mu (location-like), alpha (shape), d (shape).
Typical uses: non-normal error modeling with adjustable tail and shape behavior.
from applpy.conversion import cdf
from applpy.distributions.continuous import ErrorRV
x = ErrorRV(mu=1, alpha=2, d=1)
print(cdf(x, 1))Description: Type-II error-family distribution with tunable shape and scale.
Parameters: a (location/shape), b (scale/shape), c (shape).
Typical uses: reliability and skewed lifetime data modeling.
from applpy.conversion import cdf
from applpy.distributions.continuous import ErrorIIRV
x = ErrorIIRV(a=0, b=1, c=2)
print(cdf(x, 1))Description: Exponential distribution with constant hazard rate.
Parameters: theta (scale, positive).
Typical uses: inter-arrival times, memoryless waiting-time systems, and reliability baselines.
from applpy.conversion import cdf
from applpy.moments import mean
from applpy.distributions.continuous import ExponentialRV
x = ExponentialRV(theta=10)
print(mean(x))
print(cdf(x, 5))Description: Exponential-power (generalized normal/Laplace family) distribution.
Parameters: theta (scale, positive), kappa (shape, positive).
Typical uses: peaked or heavy/light-tailed error models.
from applpy.conversion import cdf
from applpy.distributions.continuous import ExponentialPowerRV
x = ExponentialPowerRV(theta=1, kappa=2)
print(cdf(x, 0))Description: Extreme-value (Gumbel-type) distribution for maxima/minima analysis.
Parameters: alpha (location), beta (scale).
Typical uses: block maxima, environmental extremes, and risk analysis.
from applpy.conversion import cdf
from applpy.distributions.continuous import ExtremeValueRV
x = ExtremeValueRV(alpha=0, beta=1)
print(cdf(x, 1))Description: F distribution from a ratio of scaled chi-square random variables.
Parameters: n1 (numerator degrees of freedom), n2 (denominator degrees of freedom).
Typical uses: ANOVA, nested-model comparisons, and variance-ratio testing.
from applpy.conversion import cdf
from applpy.distributions.continuous import FRV
x = FRV(n1=5, n2=10)
print(cdf(x, 1))Description: Gamma distribution for positive continuous quantities.
Parameters: theta (scale, positive), kappa (shape, positive).
Typical uses: waiting times, rainfall/claim amounts, and Bayesian priors for rates.
from applpy.moments import mean
from applpy.distributions.continuous import GammaRV
x = GammaRV(theta=2, kappa=3)
print(mean(x))Description: Generalized Pareto distribution with flexible tail behavior.
Parameters: theta (scale, positive), delta (location/threshold), kappa (shape/tail).
Typical uses: peaks-over-threshold extreme-value analysis.
from applpy.conversion import cdf
from applpy.distributions.continuous import GeneralizedParetoRV
x = GeneralizedParetoRV(theta=1, delta=0, kappa=0.2)
print(cdf(x, 2))Description: Gompertz distribution with exponentially changing hazard.
Parameters: theta (scale, positive), kappa (shape).
Typical uses: mortality and survival models.
from applpy.moments import mean
from applpy.distributions.continuous import GompertzRV
x = GompertzRV(theta=1, kappa=0.5)
print(mean(x))Description: Inverse distribution family used for skewed positive lifetimes.
Parameters: theta (scale/location), delta (shape/location), kappa (shape).
Typical uses: reliability and lifetime modeling with asymmetric tails.
from applpy.conversion import cdf
from applpy.distributions.continuous import IDBRV
x = IDBRV(theta=1, delta=0, kappa=2)
print(cdf(x, 1))Description: Inverse-gamma distribution for positive scales and variances.
Parameters: alpha (shape, positive), beta (scale, positive).
Typical uses: Bayesian priors for variance and precision-like quantities.
from applpy.moments import mean
from applpy.distributions.continuous import InverseGammaRV
x = InverseGammaRV(alpha=3, beta=2)
print(mean(x))Description: Inverse Gaussian (Wald) distribution for first-passage times.
Parameters: theta (scale, positive), mu (mean/location, positive).
Typical uses: diffusion hitting times and positive skewed duration data.
from applpy.conversion import cdf
from applpy.distributions.continuous import InverseGaussianRV
x = InverseGaussianRV(theta=2, mu=1)
print(cdf(x, 1))Description: Kolmogorov-Smirnov related distribution for KS statistics.
Parameters: n (sample size, positive integer).
Typical uses: modeling the finite-sample KS test statistic distribution.
from applpy.conversion import cdf
from applpy.distributions.continuous import KSRV
x = KSRV(n=20)
print(cdf(x, 0.2))Description: Laplace (double exponential) distribution with sharp center and heavier tails than normal.
Parameters: omega (scale, positive), theta (location).
Typical uses: robust error modeling and L1-style noise assumptions.
from applpy.conversion import cdf
from applpy.distributions.continuous import LaPlaceRV
x = LaPlaceRV(omega=1, theta=0)
print(cdf(x, 0))Description: Log-gamma distribution on positive support with flexible skew.
Parameters: alpha (shape, positive), beta (scale/rate, positive).
Typical uses: skewed positive data and transformed gamma-type models.
from applpy.moments import mean
from applpy.distributions.continuous import LogGammaRV
x = LogGammaRV(alpha=2, beta=1)
print(mean(x))Description: Logistic distribution with sigmoid CDF and moderate tails.
Parameters: kappa (scale, positive), theta (location).
Typical uses: latent-variable models and growth/response curves.
from applpy.conversion import cdf
from applpy.distributions.continuous import LogisticRV
x = LogisticRV(kappa=1, theta=0)
print(cdf(x, 0))Description: Log-logistic distribution with positive support and heavy tails.
Parameters: theta (scale, positive), kappa (shape, positive).
Typical uses: survival/reliability modeling and hazard functions with non-monotonic behavior.
from applpy.conversion import cdf
from applpy.distributions.continuous import LogLogisticRV
x = LogLogisticRV(theta=1, kappa=2)
print(cdf(x, 1))Description: Log-normal distribution for multiplicative positive processes.
Parameters: mu (log-location), sigma (log-scale, positive).
Typical uses: finance, biological growth, and right-skewed duration/size data.
from applpy.moments import mean
from applpy.distributions.continuous import LogNormalRV
x = LogNormalRV(mu=0, sigma=1)
print(mean(x))Description: Lomax (Pareto Type II) heavy-tailed positive distribution.
Parameters: kappa (shape, positive), theta (scale, positive).
Typical uses: claim severity, internet traffic, and heavy-tail reliability data.
from applpy.conversion import cdf
from applpy.distributions.continuous import LomaxRV
x = LomaxRV(kappa=2, theta=1)
print(cdf(x, 2))Description: Makeham survival distribution with age-dependent and constant hazard terms.
Parameters: theta (scale, positive), delta (positive shape), kappa (shape).
Typical uses: actuarial mortality and lifetime risk decomposition.
from applpy.conversion import cdf
from applpy.distributions.continuous import MakehamRV
x = MakehamRV(theta=1, delta=1, kappa=0.2)
print(cdf(x, 1))Description: Muth distribution for right-skewed positive outcomes.
Parameters: kappa (shape, positive).
Typical uses: reliability and response-time modeling.
from applpy.conversion import cdf
from applpy.distributions.continuous import MuthRV
x = MuthRV(kappa=1)
print(cdf(x, 1))Description: Normal (Gaussian) distribution.
Parameters: mu (mean/location), sigma (standard deviation, positive).
Typical uses: measurement noise, CLT approximations, and baseline parametric modeling.
from applpy.conversion import cdf
from applpy.moments import mean
from applpy.distributions.continuous import NormalRV
x = NormalRV(mu=0, sigma=1)
print(mean(x))
print(cdf(x, 0))Description: Pareto heavy-tail distribution on positive support.
Parameters: theta (scale/minimum, positive), kappa (shape/tail index, positive).
Typical uses: wealth modeling, file-size distributions, and tail-risk studies.
from applpy.conversion import cdf
from applpy.distributions.continuous import ParetoRV
x = ParetoRV(theta=1, kappa=2)
print(cdf(x, 2))Description: Rayleigh distribution for magnitudes of 2D normal vectors.
Parameters: theta (scale, positive).
Typical uses: signal processing, wind-speed approximations, and random vector magnitudes.
from applpy.moments import mean
from applpy.distributions.continuous import RayleighRV
x = RayleighRV(theta=2)
print(mean(x))Description: Student's t distribution with heavier tails than normal.
Parameters: N (degrees of freedom).
Typical uses: small-sample inference and robust mean modeling.
from applpy.conversion import cdf
from applpy.distributions.continuous import TRV
x = TRV(N=10)
print(cdf(x, 0))Description: Triangular distribution over a bounded interval with a mode.
Parameters: a (lower bound), b (mode), c (upper bound).
Typical uses: project planning and bounded expert-estimate inputs.
from applpy.moments import mean
from applpy.distributions.continuous import TriangularRV
x = TriangularRV(a=0, b=2, c=5)
print(mean(x))Description: Continuous uniform distribution over [a, b].
Parameters: a (lower bound), b (upper bound).
Typical uses: non-informative bounded models and simulation primitives.
from applpy.conversion import cdf
from applpy.distributions.continuous import UniformRV
x = UniformRV(a=0, b=1)
print(cdf(x, 0.25))Description: Weibull distribution with flexible monotone hazard behavior.
Parameters: theta (scale, positive), kappa (shape, positive).
Typical uses: reliability engineering, failure-time analysis, and survival modeling.
from applpy.moments import mean
from applpy.distributions.continuous import WeibullRV
x = WeibullRV(theta=2, kappa=1.5)
print(mean(x))