An enormous and important theoretical effort has been directed at studying the origin of broad-tailed probability distribution functions observed for numerous physical quantities measured in fluid turbulence. Despite the amount of attention this problem has received, there are still few rigorous results. One model which has been amenable to rigorous analysis is the Majda model for the diffusion of a passive scalar in the presence of a random, rapidly fluctuating linear shear layer, an anisotropic analog of the Kraichnan model. Previous work, by Majda, lead to explicit formulas for the moments of the distribution of the scalar. We examine this model, and construct the explicit large moment number asymptotics. Using properties of entire functions of finite order, we calculate the rigorous tail of the limiting probability distribution function for normalized scalar fluctuations. Through this process, we obtain an explicit relation between the limiting tail of the scalar probability distribution function and that of the scalar gradient. We additionally apply the method to moments derived asymptotically by Son, and those derived phenomenologically by She and Orszag.