Real and Complex Analysis of Common Probability Density Functions

Abstract

In this paper, the most common method of probability density function integration Gaussian Integration is analyzed in real and complex methods. This paper begins by clarifying the meaning and significance of integrating the probability density function. After this, it elucidates and derive the mathematical methods for the analysis of integrals, namely the multiple integral method in real analysis and the residue theorem method in complex analysis. At the same time, the paper also makes it clear to analyze the most typical Gaussian integral in the probability density function. Since then, this integration method has been used respectively for real analysis and complex analysis. Different means of analysis including Pinch criterion or Residue theorem have been applied comprehensively. In this session, different forms and formats of Gaussian integrals are explored to a certain extent. Finally, the benefits of complex analysis and real analysis for the synthesis of probability density function including Gaussian integral are summarized, and the possible applications and prospects of this new analysis method are prospected. However, the limitations of the current approach have also been mentioned.