The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the
Also, a
That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.