6. Appendices

6.1 Volumetric Probability and Probability Density A probability distribution over a manifold can be represented by a volumetric probability F (x), defined through P (A) = ∫A dV (x) F (x) , 6.1 or by a probability density f (x), defined through P (A) = ∫A d x ƒ (x) , 6.2 where dx = dx1dx2…. While, under a change of variables, a probability density behaves as a density (i.e., its value at a point gets multiplied by the Jacobian of the transformation), a volumetric probability is a scalar (i.e., its value at a point remains invariant: it is defined independently of any coordinate system). Defining the volume density through V (A) = ∫A d x υ (x) 6.3 and considering the expression we obtain dV (x) =υ (x) dx . 6.4 It follows that the relation between volumetric probability and probability density is ƒ (x) =υ (x) F (x) . 6.5 While the homogeneous probability distribution (the one assigning equal probabilities to equal volumes of the space) is, in general, not represented by a constant probability density, it is always represented by a constant volumetric probability. Although I prefer, in my own work, to use volumetric probabilities, I have chosen in this text to use probability densities (for pedagogical reasons).