  Bowen's formula relates the Hausdorff dimension of a conformal repeller to
the zero of a `pressure' function. We present an elementary, self-contained
proof which bypasses measure theory and the Thermodynamic Formalism to show
that Bowen's formula holds for $C^1$ conformal repellers. We consider
time-dependent conformal repellers obtained as invariant subsets for sequences
of conformally expanding maps within a suitable class. We show that Bowen's
formula generalizes to such a repeller and that if the sequence is picked at
random then the Hausdorff dimension of the repeller almost surely agrees with
its upper and lower Box dimensions and is given by a natural generalization of
Bowen's formula. For a random uniformly hyperbolic Julia set on the Riemann
sphere we show that if the family of maps and the probability law depend
real-analytically on parameters then so does its almost sure Hausdorff
dimension.
