正态云模型的重尾性质证明

正态分布和重尾分布在概率研究中具有非常重要的地位，二者具有完全不同的数学形式和物理意义。正态分布的密度函数以指数函数衰减至0，服从正态分布的随机变量，其绝大多数取值在其期望附近，偏离期望很大的取值很少。而服从重尾分布的随机变量，其尾分布函数具有重尾特性，密度函数以幂指数衰减至0。笔者证明了正态云模型是具有均值的重尾分布，是介于正态分布与重尾分布之间的中间状态，正态云模型的参数超熵He是可以实现正态分布向重尾分布转换的桥梁。

Proof of the heavy tailed property of normal cloud model

Normal distribution and heavy tailed distribution are very important in probability theories. They have totally different mathematical forms and physical meanings. The probability density function of normal distribution decay exponentially to 0. The majority of normal random variable values are around the mathematical expectation. The tailed distribution function of the random variables that obey heavy tailed distribution shows heavy tailed characteristic. The probability density function decays power exponentially to 0.In this paper, we proved that the normal cloud model is heavy tailed distribution and its mathematical expectation exists.It is intermediate between notmal distribution and heavy tailed distribution. The parameter He(hyper entropy) of the normal cloud model is the bridge from normal distribution to heavy tailed distribution.