We consider random products of $SL(2, \mathbb{R})$ matrices that depend on a
parameter in a non-uniformly hyperbolic regime. We show that if the dependence
on the parameter is monotone then almost surely the random product has upper
(limsup) Lyapunov exponent that is equal to the value prescribed by the
Furstenberg Theorem (and hence positive) for all parameters, but the lower
(liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of
parameters of zero Hausdorff dimension. As a byproduct of our methods, we
provide a purely geometrical proof of Spectral Anderson Localization for
discrete Schr\"odinger operators with random potentials (including the
Anderson-Bernoulli model) on a one dimensional lattice.