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
)
=
1
[
1
−
I
(
f
<
0
)
−
sgn
(
f
)
Φ
(
0
,
m
,
s
)
]
2
π
s
2
exp
{
−
1
2
s
2
(
y
f
f
−
m
)
2
}
{\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.
Special cases
ƒ = 1 gives a truncated normal distribution.