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We study the initiation and evolution of aneurysmal morphology in a pressurized soft tube where the elastic modulus is non-uniform in the radial direction. The primary deformation prior to instability is characterized within the framework of nonlinear elasticity for a general material constitution and a generic modulus gradient. To unravel the influence of modulus gradient on aneurysm formation, we employ the incompressible Gent model and select three representative modulus gradients, including a linear, an exponential, and a sinusoidal function. In particular, the sinusoidal distribution can be used to model actual artery structure. In addition, two prototypical loading conditions are considered, namely, either the resultant axial force or the axial length can be fixed. Based on an explicit bifurcation condition in terms of the internal pressure and the resultant axial force for aneurysm formation or localized bulging, an exhaustive theoretical analysis on bulge initiation is carried out and the effect of geometric and material parameters and modulus gradient on the critical stretch generating localized bulging is revealed. It turns out that the modulus mismatch, as well as the position of maximum modulus, can dramatically affect the onset of localized bulging. Then we analytically elucidate the influence of modulus gradient on bulge propagation and conduct a finite element analysis of bulge evolution based on a robust finite element model established in Abaqus by UHYPER subroutine coding. Interestingly, it is found that a sinusoidally distributed modulus has negligible influence on the critical stretch of bulge initiation, the deformation process of bugle growth, and the maximum size of a bulge. The current analysis can provide useful insight into the biological evolution of human artery and into localized instabilities in graded structures.