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

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

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.

• ƒ = 1 gives a truncated normal distribution.

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

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