Probabilistic fracture analysis is performed for predicting uncertain fracture responses of Functionally Graded Material (FGM) structures. The uncertainties in material properties including Young's modulus and fracture toughness are considered. The limit state function for a crack initiation event is defined in terms of the J-integral for FGMs. The First-Order-Reliability- Method (FORM) is used in conjunction with a finite element code that computes the J-integral with high accuracy. A two-step probabilistic analysis procedure is proposed to investigate the effects of the uncertainties in the spatial distribution of Young's modulus on the probability of crack initiation in FGMs. First, we investigate the effects of the uncertainties in the shape of the spatial distribution by considering the slope and the location of the inflection point of a spatial distribution profile as random quantities. Second, we investigate the effects of the spatial fluctuations of Young's modulus by making use of a discretized random field. The companion paper (Part II) implements this method into a finite element fracture analysis code and presents numerical examples.