Box–Cox distribution

In statistics, the Box–Cox distribution (also known as the power-normal distribution) is the distribution of a random variable X for which the Box–Cox transformation on X follows a truncated normal distribution. It is a continuous probability distribution having probability density function (pdf) given by

f(y)={\frac {1}{\left(1-I(f<0)-\operatorname {sgn}(f)\Phi (0,m,{\sqrt {s}})\right){\sqrt {2\pi s^{2}}}}}\exp \left\{-{\frac {1}{2s^{2}}}\left({\frac {y^{f}}{f}}-m\right)^{2}\right\}

for y > 0, where m is the location parameter of the distribution, s is the dispersion, ƒ is the family parameter, I is the indicator function, Φ is the cumulative distribution function of the standard normal distribution, and sgn is the sign function.

Special cases

ƒ = 1 gives a truncated normal distribution.

References

Freeman, Jade; Reza Modarres. "Properties of the Power-Normal Distribution" (PDF). U.S. Environmental Protection Agency.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
multinomial
- Dirichlet
- negative
Continuous:
Dirichlet
- generalized
multivariate Laplace
multivariate normal
multivariate stable
multivariate t
normal-gamma
- inverse
Matrix-valued:
LKJ
matrix normal
matrix t
matrix gamma
- inverse
Wishart
- normal
- inverse
- normal-inverse
- complex

Univariate (circular) directional: Circular uniform; univariate von Mises; wrapped normal; wrapped Cauchy; wrapped exponential; wrapped asymmetric Laplace; wrapped Lévy
Bivariate (spherical): Kent
Bivariate (toroidal): bivariate von Mises
Multivariate: von Mises–Fisher; Bingham

Degenerate
and singular

Degenerate: Dirac delta function
Singular: Cantor

Families

Category
Commons

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