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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.