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In this work we use a new approach to model error events in long-haul optical fiber transmission systems. Existing approaches for obtaining probability density functions (PDFs) rely on numerical simulations or analytical approximations. Numerical simulations make far tails of the PDFs difficult to obtain, while analytical approximations are often inaccurate, as they neglect nonlinear interaction between pulses and noise. Our approach combines the instanton method from statistical mechanics, to model far tails of the PDFs, with numerical simulations to refine the middle part of the PDFs. We combine the two methods by using an orthogonal polynomial expansion constructed specifically for this problem. We demonstrate the approach on an example of a specific submarine transmission system.