  We develop a new variational scheme to approximate the position dependent
spatial probability distribution of a zero dimensional manifold in a random
medium. This celebrated 'toy-model' is associated via a mapping with directed
polymers in 1+1 dimension, and also describes features of the
commensurate-incommensurate phase transition. It consists of a pointlike
'interface' in one dimension subject to a combination of a harmonic potential
plus a random potential with long range spatial correlations. The variational
approach we develop gives far better results for the tail of the spatial
distribution than the hamiltonian version, developed by Mezard and Parisi, as
compared with numerical simulations for a range of temperatures. This is
because the variational parameters are determined as functions of position. The
replica method is utilized, and solutions for the variational parameters are
presented. In this paper we limit ourselves to the replica symmetric solution.
