<![CDATA[Probability distributions serve as the cornerstone of inferential statistics, yet the mathematical proofs of their genealogical relationships are frequently presented fragmentedly using inconsistent methodologies in academic literature. This study aims to reconstruct and unify the transformation pathways of major continuous probability distributions—comprising Standard Normal, Chi-Square, Gamma, Beta , F-Snedecor, and t-Student into a single, cohesive theoretical framework. The analytical derivation relies on the rigorous application of multistage bivariate Jacobian matrix transformations and asymptotic analysis, thereby eliminating dependency on Moment Generating Functions (MGF) which often fail to converge within specific parameter spaces. The results demonstrate that this transformation chain does not merely form a one-way linear trajectory, but rather a cyclic (circular) conceptual structure. The Standard Normal distribution acts as the origin that derives Chi-Square, Gamma, Beta, and F, before culminating in the t-distribution. Furthermore, through asymptotic limit proofs as the degrees of freedom approach infinity, the probability density function of the t-Student distribution mathematically converges back into the Standard Normal distribution, perfectly closing the genealogical loop. The theoretical contribution of this study clarifies the underlying mathematical assumptions, eliminating the "black-box" perspective on t-tests, ANOVA, and Chi-Square tests, while providing a computational foundation for optimizing random number generator algorithms in stochastic simulations.]]>
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