We consider words \(\pi_1\pi_2\pi_3\ldots\pi_n\) of length \(n\), where \(\pi_i \in \mathbb{N}\) are independently generated with a geometric probability
\[P({\pi} = k) = p(q)^{k-1} \text{where p + q = 1}. \]
Let \(d\) be a fixed non-negative integer. We say that we have an ascent of size \(d\) or more, an ascent of size less than \(d\), a level, and a descent if \({\pi}_{i+1} \geq {\pi}_i+d \), \({\pi}_{i+1} {\pi}_{i+1} \), respectively.We determine the mean and variance of the number of ascents of size less than \(d\) in a random geometrically distributed word. We also show that the distribution is Gaussian as \(n\) tends to infinity.