Distributions Guide

This section provides detailed information about all probability distributions available in greybox.

Overview

greybox provides a comprehensive set of probability distributions for modeling different types of data. Each distribution family includes:

Density function (d*) - Probability density function (PDF)
Cumulative distribution function (p*) - CDF
Quantile function (q*) - Inverse CDF
Random generator (r*) - Random sample generation

Common Parameters

Most distribution functions share common parameters:

q or x : Value(s) at which to evaluate the function
loc or mu : Location parameter (often the mean)
scale : Scale parameter (often the standard deviation)
shape : Shape parameter(s) specific to the distribution
log / log_p : If True, return the log of the probability
lower_tail : If True (default), probabilities are P[X <= x]

Continuous Univariate Distributions

Normal (Gaussian)

dnorm(q, mean=0, sd=1, log=False)

Normal (Gaussian) distribution density.

Parameters:

q – Value(s) at which to evaluate
mean – Mean (location parameter)
sd – Standard deviation (scale parameter)
log – If True, return log density

Returns:

Density at q

Example:

from greybox import dnorm
dnorm(0, mean=0, sd=1)  # ~0.3989

Laplace

dlaplace(q, loc=0, scale=1, log=False)

Laplace (double exponential) distribution density.

Parameters:

q – Value(s) at which to evaluate
loc – Location parameter (median)
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

S Distribution

ds(q, mu=0, scale=1, log=False)

S-distribution density - a heavy-tailed distribution.

Parameters:

q – Value(s) at which to evaluate
mu – Location parameter
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Generalized Normal

dgnorm(q, mu=0, scale=1, shape=1, log=False)

Generalized Normal distribution density.

Parameters:

q – Value(s) at which to evaluate
mu – Location parameter
scale – Scale parameter
shape – Shape parameter (controls tail weight) - shape=1: Laplace - shape=2: Normal - shape<2: Heavy tails - shape>2: Light tails
log – If True, return log density

Returns:

Density at q

Example:

from greybox import dgnorm
# Normal-like
dgnorm(0, mu=0, scale=1, shape=2)
# Heavy-tailed
dgnorm(5, mu=0, scale=1, shape=1)

Logistic

dlogis(q, loc=0, scale=1, log=False)

Logistic distribution density.

Parameters:

q – Value(s) at which to evaluate
loc – Location parameter
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Student’s t

dt(q, df, loc=0, scale=1, log=False)

Student’s t distribution density.

Parameters:

q – Value(s) at which to evaluate
df – Degrees of freedom
loc – Location parameter
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Asymmetric Laplace

dalaplace(q, mu=0, scale=1, alpha=0.5, log=False)

Asymmetric Laplace distribution density for quantile regression.

Parameters:

q – Value(s) at which to evaluate
mu – Location parameter
scale – Scale parameter
alpha – Asymmetry parameter (0 < alpha < 1) - alpha < 0.5: Right-skewed - alpha > 0.5: Left-skewed
log – If True, return log density

Returns:

Density at q

Log-Transformed Distributions

These distributions model the log of the response variable.

Log-Normal

dlnorm(q, meanlog=0, sdlog=1, log=False)

Log-Normal distribution density.

Parameters:

q – Value(s) - must be positive
meanlog – Mean of the underlying normal distribution
sdlog – Standard deviation of the underlying normal distribution
log – If True, return log density

Returns:

Density at q

Note: For positive-valued data that is right-skewed.

Log-Laplace

dllaplace(q, loc=0, scale=1, log=False)

Log-Laplace distribution density.

Parameters:

q – Value(s) - must be positive
loc – Location parameter (of log values)
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Log-S

dls(q, loc=0, scale=1, log=False)

Log-S distribution density.

Parameters:

q – Value(s) - must be positive
loc – Location parameter (of log values)
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Log-Generalized Normal

dlgnorm(q, mu=0, scale=1, shape=2, log=False)

Log-Generalized Normal distribution density.

Parameters:

q – Value(s) - must be positive
mu – Location parameter (of log values)
scale – Scale parameter
shape – Shape parameter
log – If True, return log density

Returns:

Density at q

Box-Cox Normal

dbcnorm(q, mu=0, sigma=1, lambda_bc=0, log=False)

Box-Cox Normal distribution density.

Parameters:

q – Value(s) - must be positive
mu – Location parameter
sigma – Scale parameter
lambda_bc – Box-Cox transformation parameter
log – If True, return log density

Returns:

Density at q

Folded and Rectified Distributions

These distributions model positive-valued data with a point mass at zero.

Folded Normal

dfnorm(q, mu=0, sigma=1, log=False)

Folded Normal distribution density.

Parameters:

q – Value(s) - must be non-negative
mu – Mean of underlying normal
sigma – Standard deviation of underlying normal
log – If True, return log density

Returns:

Density at q

Rectified Normal

drectnorm(q, mu=0, sigma=1, log=False)

Rectified Normal distribution density.

Parameters:

q – Value(s) - must be non-negative
mu – Mean of underlying normal
sigma – Standard deviation of underlying normal
log – If True, return log density

Returns:

Density at q

Distributions for Positive Values

Inverse Gaussian

dinvgauss(q, mu=1, scale=1, log=False)

Inverse Gaussian distribution density.

Parameters:

q – Value(s) - must be positive
mu – Mean parameter
scale – Scale parameter
log – If True, return log density

Returns:

Density at q

Gamma

dgamma(q, shape=1, scale=1, log=False)

Gamma distribution density.

Parameters:

q – Value(s) - must be positive
shape – Shape parameter (often denoted alpha)
scale – Scale parameter (often denoted theta)
log – If True, return log density

Returns:

Density at q

Exponential

dexp(q, loc=0, scale=1, log=False)

Exponential distribution density.

Parameters:

q – Value(s) - must be non-negative
loc – Location parameter
scale – Scale parameter (1/rate)
log – If True, return log density

Returns:

Density at q

Chi-Squared

dchi2(q, df, log=False)

Chi-squared distribution density.

Parameters:

q – Value(s) - must be non-negative
df – Degrees of freedom
log – If True, return log density

Returns:

Density at q

Count Distributions

Poisson

dpois(q, mu, log=False)

Poisson distribution probability mass function.

Parameters:

q – Value(s) - must be non-negative integers
mu – Mean/lambda parameter
log – If True, return log probability

Returns:

Probability mass at q

Negative Binomial

dnbinom(q, mu=1, size=1, log=False)

Negative Binomial distribution probability mass function.

Parameters:

q – Value(s) - must be non-negative integers
mu – Mean parameter
size – Dispersion parameter
log – If True, return log probability

Returns:

Probability mass at q

Geometric

dgeom(q, prob, log=False)

Geometric distribution probability mass function.

Parameters:

q – Value(s) - must be non-negative integers
prob – Probability of success
log – If True, return log probability

Returns:

Probability mass at q

Binary/Bounded Distributions

Binomial

dbinom(q, size, prob, log=False)

Binomial distribution probability mass function.

Parameters:

q – Value(s) - must be integers between 0 and size
size – Number of trials
prob – Probability of success
log – If True, return log probability

Returns:

Probability mass at q

Beta

dbeta(q, a, b, log=False)

Beta distribution density.

Parameters:

q – Value(s) - must be in [0, 1]
a – First shape parameter (alpha)
b – Second shape parameter (beta)
log – If True, return log density

Returns:

Density at q

Logit-Normal

dlogitnorm(q, mu=0, sigma=1, log=False)

Logit-Normal distribution density.

Parameters:

q – Value(s) - must be in (0, 1)
mu – Mean of underlying normal (logit scale)
sigma – Standard deviation (logit scale)
log – If True, return log density

Returns:

Density at q

CDF-Based Distributions

These distributions use the CDF in the likelihood.

Logistic CDF

plogis(q, location=0, scale=1, log_p=False, lower_tail=True): Logistic cumulative distribution function.

Probit (Normal CDF)

pnorm(q, loc=0, scale=1, log_p=False, lower_tail=True): Normal cumulative distribution function.

Summary

greybox supports the following distributions. For each distribution family, use the appropriate prefix:

d* - Probability density/mass function (PDF/PMF)
p* - Cumulative distribution function (CDF)
q* - Quantile function (inverse CDF)
r* - Random number generation

The choice of distribution depends on the nature of your data:

Continuous, symmetric: Normal, Laplace, Logistic, Student’s t
Heavy tails: Laplace, S, Generalized Normal, Student’s t
Positive, right-skewed: Log-Normal, Gamma, Inverse Gaussian
Count data: Poisson, Negative Binomial, Geometric
Proportions: Beta, Logit-Normal, Binomial
Zero-inflated: Folded Normal, Rectified Normal