  The maximal point of the Airy2 process minus a parabola is believed to
describe the scaling limit of the end-point of the directed polymer in a random
medium, which was proved to be true for a few specific cases. Recently two
different formulas for the joint distribution of the location and the height of
this maximal point were obtained, one by Moreno Flores, Quastel and Remenik,
and the other by Schehr. The first formula is given in terms of the Airy
function and an associated operator, and the second formula is expressed in
terms of the Lax pair equations of the Painleve II equation. We give a direct
proof that these two formulas are the same.
