Abstract: The Morozov'’s discrepancy principle has been used as a general criterion to compute the regularization parameter in inverse problems. The Morozov'’s principle is established when the error in the measurements is assumed to have a Gaussian probability density function, with zero mean and \sigam^2 variance. The goal of this paper is to present a generalized discrepancy principle for distributions in which the second moment is not defined. The generalized discrepancy principle is applied to several distributions: uniform, Gaussian, Cauchy, t-Student, Tsallis. The estimation of the initial condition in a heat transport problem is used as a test-problem.