We derive the exact probability density function of the signal to noise ratio (SNR) impairment caused by polarization dependent loss (PDL) in coherent optical systems. Using thehinge model for PDL statistics, we construct a sequence of variable transformations that yields the expression of the SNR probability density function by means of a sequence of nested recursive integrals. Monte Carlo simulations validate the analytical formulation assuming realistic PDL levels and link configurations, showing excellent agreement. The model accurately captures SNR variability introduced by cascaded PDL elements and provides a foundation for statistical margin assessment, digital twin modeling, and quality of transmission estimation in flexible, low marginoptical networks.