New Results for Pearson Type III Family of Distributions and Application in Wireless Power Transfer

The Pearson type III and the log Pearson type III distributions have been considered in several scientific fields, as in hydrology and seismology. In this article, we present new results for these distributions and we utilize them, for the first time in the literature, to investigate the statistical behavior of wireless power transfer, which can prolong the lifetime of Internet of Things networks, considering the nonlinear relationship between the received and harvested power, which can be precisely modeled by using the logistic function. Specifically, we present new closed-form expressions for the statistical properties of a general form of the Pearson type III and the log Pearson type III distributions and we utilize them to introduce a new member of the Pearson type III family, the logit Pearson type III distribution, through which the logit gamma and the logit exponential distributions are also defined. Moreover, we derive closed-form expressions for the probability density function, the cumulative distribution function and moments of the distributions of the sum, the log sum, and the logit sum of Pearson type III random variables. Furthermore, taking into account that the Pearson type III family of distributions is closely related to the considered nonlinear energy harvesting model the statistical properties of the distribution of the harvested power are derived, for both single input single output and multiple input single output scenarios with or without channel state information at the transmitter.