We investigated the statistical properties of the Moran random walk $(Y_n{)}_n$ in one dimension, focusing on short memory. Specifically, employing generating function techniques, we determined the cumulative distribution function and the mean of the height $H_n$. Furthermore, we derived explicit expressions for the distribution, mean, and variance of $Y_n$, along with its asymptotic distribution. Finally, we provided the distribution of the waiting time ${\tau }_h$, which represents the number of steps required to reach a specified level $h$, as the conclusion of our study.