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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
In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions.It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association (such as the Pearson correlation between variables), and for other data stabilization procedures.
Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function.
The parameter q represents the degree of non-extensivity of the distribution. ... proposed by George Box and David Cox in 1964. [2] q-exponential
Since the power transformation family also includes the identity transformation, this approach can also indicate whether it would be best to analyze the data without a transformation. In regression analysis, this approach is known as the Box–Cox transformation.
The exponential distribution is recovered as . Originally proposed by the statisticians George Box and David Cox in 1964, [2] and known as the reverse Box–Cox transformation for =, a particular case of power transform in statistics.
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The basic form as given by Box and Muller takes two samples from the uniform distribution on the interval (0,1) and maps them to two standard, normally distributed samples. The polar form takes two samples from a different interval, [−1,+1], and maps them to two normally distributed samples without the use of sine or cosine functions.