A unified statistical and phenomenological approach to geometrization of classical thermodynamics is proposed. It is shown that any r-parameter probability distribution function leads to a Riemannian metric of the parameter space with components of the metric tensor represented by fluctuations of the associated stochastic variables. A general theory of conformal transformations relating metrics induced by various distribution functions and connected with various thermodynamic potentials is developed within the framework of the contact geometry of phenomenological thermodynamics. Relations between Weihold's and Ruppeiner's metrics are further clarified. The theory is illustrated by the P– T and μ– T distribution functions.