We study probability measures defined by the variation of the sum of digits in the Zeckendorf representation. For r ≥ 0 and d ∈ Z, we consider μ^(r)(d), the density of integers n ∈ N for which the sum of digits increases by d when r is added to n. We give a probabilistic interpretation of μ^(r) via the dynamical system provided by the odometer of Zeckendorf-adic integers and its unique invariant measure. We give an algorithm for computing μ^(r) and we prove the exponential decay of μ^(r)(d) as d → -∞, as well as the formula μ^(F_ℓ) = μ⁽¹⁾ where F_ℓ is a term of the Fibonacci sequence. Finally, we decompose the Zeckendorf representation of an integer r into so-called "blocks" and show that when added to an adic Zeckendorf integer, the successive actions of these blocks can be seen as a sequence of mixing random variables.