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Anderson localization is the ubiquitous phenomenon of inhibition of transport of classical and quantum waves in a disordered medium. In dimension one, it is well known that all states are localized, implying that the distribution of an initially narrow wave-packet released in a disordered potential will, at long time, decay exponentially on the scale of the localization length. However, the exact shape of the stationary localized distribution differs from a purely exponential profile and has been computed almost fifty years ago by Gogolin. Using the atomic quantum kicked rotor, a paradigmatic quantum simulator of Anderson localization physics, we study this asymptotic distribution by two complementary approaches. First, we discuss the connection of the statistical properties of the system's localized eigenfunctions and their exponential decay with the localization length of the Gogolin distribution. Next, we make use of our experimental platform, realizing an ideal Floquet disordered system, to measure the long-time probability distribution and highlight the very good agreement with the analytical prediction compared to the purely exponential one over 3 orders of magnitude.