The instantaneous unit hydrograph is conceived as a random function of climate and geomorphology varying with the characteristics of the rainfall excess. The probability density functions of the peak and time to peak of the IUH are analytically derived as functions of the rainfall characteristics and the basin geomorphological parameters. The main characteristics of these pdf's are studied, and a new approach to hydrologic similarity is initiated under the concept of the geomorphoclimatic IUH. For a given set of geomorphologic characteristics and a particular intensity and duration of rainfall, the peak and time to peak of the IUH corresponding to those values can be easily estimated.