In this work, we address the recent experiments [C. Altimiras et al., Nat. Phys. 6, 34 (2010); H. le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010);C. Altimiras et al., Phys. Rev. Lett. 105, 226804 (2010)], where an electron distribution function at the quantum Hall (QH) edge at filling factor ν=2 has been measured with high precision. It has been reported that the energy of electrons injected into one of the two chiral edge channels with the help of a quantum point contact (QPC) is equally distributed between them, in agreement with earlier predictions, one being based on the Fermi gas approach [A. M. Lunde et al., Phys. Rev. B 81, 041311(R) (2010)] and the other utilizing the Luttinger-liquid theory [P. Degiovanni et al., Phys. Rev. B 81, 121302(R) (2010)]. We argue that the physics of the energy relaxation process at the QH edge may in fact be more rich, providing the possibility for discriminating between two physical pictures in experiment. Namely, using the recently proposed nonequilibrium bosonization technique [I. P. Levkivskyi et al., Phys. Rev. Lett., 103, 036801 (2009)], we evaluate the electron distribution function and find that the initial “double-step” distribution created at a QPC evolves through several intermediate asymptotics before reaching eventual equilibrium state. At short distances, the distribution function is found to be asymmetric due to non-Gaussian current noise effects. At larger distances, where noise becomes Gaussian, the distribution function acquires symmetric Lorentzian shape. Importantly, in the regime of low QPC transparencies T, the width of the Lorentzian scales linearly with T, in contrast to the case of equilibrium Fermi distribution, the width of which scales as T−−√. Therefore, we propose to do measurements at low QPC transparencies. We suggest that the missing energy paradox may be explained by the nonlinearities in the spectrum of edge states.