  We outline basic properties of a symmetric random walk in one dimension, in
which the length of the nth step equals lambda^n, with lambda<1. As the number
of steps N-->oo, the probability that the endpoint is at x, P_{lambda}(x;N),
approaches a limiting distribution P_{lambda}(x) that has many beautiful
features. For lambda<1/2, the support of P_{lambda}(x) is a Cantor set. For
1/2<=lambda<1, there is a countably infinite set of lambda values for which
P_{lambda}(x) is singular, while P_{lambda}(x) is smooth for almost all other
lambda values. In the most interesting case of lambda=(sqrt{5}-1)/2=g, P_g(x)
is riddled with singularities and is strikingly self-similar. The
self-similarity is exploited to derive a simple form for the probability
measure M(a,b)= int_a^b P_g(x) dx.
