  In previous work Majda and McLaughlin computed explicit expressions for the
$2N$th moments of a passive scalar advected by a linear shear flow in the form
of an integral over ${\bf R}^N$. In this paper we first compute the asymptotics
of these moments for large moment number. We are able to use this information
about the large N behavior of the moments, along with some basic facts about
entire functions of finite order, to compute the asymptotics of the tails of
the probability distribution function. We find that the probability
distribution has Gaussian tails when the energy is concentrated in the largest
scales. As the initial energy is moved to smaller and smaller scales we find
that the tails of the distribution grow longer, and the distribution moves
smoothly from Gaussian through exponential and ``stretched exponential''. We
also show that the derivatives of the scalar are increasingly intermittent, in
agreement with experimental observations, and relate the exponents of the
scalar derivative to the exponents of the scalar.
