In recent work, Mansour [Discrete Math. Theoret. Computer Science 11, 2009, 173--186] considers the problem of enumerating all words of length n over {1,2,...,k} (where k is a given integer), that have the total variation equal to a given integer m. In the present paper we study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities. We are interested in the probabilities of words of given type to have a given total variation.