We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P(H,t) of the height H at a given point x takes the scaling form P(H,t) ∼(-Φ_ drop(H)/√(t)) where the rate function Φ_ drop(H) is computed exactly. While it is Gaussian in the center, i.e., for small H, the PDF has highly asymmetric non-Gaussian tails which we characterize in detail. This function Φ_ drop(H) is

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... surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite size models belonging to the KPZ universality class. Thanks to a recently discovered connection between KPZ and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full couting statistics at the edge.