A Consideration on a Family of q-Normal Distributions

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In this paper, the noble probability density function, called the q-normal distribution, is given by considering Tsallis entropy which is one of the generalized entropies, the first q-moment and the second q-moment, where the q-moment is defined by the moment evaluated by the escort distribution. The q-normal distribution determines the specific distribution when the parameter q and its mean and variance are given. This q-normal distribution can smoothly connect the various probability density functions. According as the parameter q changes from - ∞ to 3, the q-normal distribution changes from the uniform distribution with compact support to the uniform distribution with non-compact support, which height is completely 0, via Cauchy distribution, t-distribution and the normal distribution etc. ft is shown that the various properties of the q-normal distribution and the duality between the q-normal distribution and its associated escort distribution. This duality relation reveals that the q-normal distribution can be related with the other q-normal distribution. Of course, there exists other distribution family to which the normal distribution belongs. However there is the meaningful difference between such other distribution family and the q-normal distributions. The most important property of the q-normal distributions is that every q-normal distribution has its own entropy subjected to the parameter q and maximizes that entropy, but the other distribution family doesn't have this fundamental property. The q-Gabor wavelet is also considered to show the prominent characters of the q-normal distribution.
社団法人電子情報通信学会の論文
2002-02-01

A Consideration on a Family of q-Normal Distributions

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