Translation lies at the heart of every living organism, it is the process synthesizing the proteins, a main component in the cells. Earlier studies have considered protein synthesis as mainly initiation limited, modeling it as a first-order reaction with respect to free ribosomes and mRNA. In such studies, the noise is therefore coming from initiation phase only. We have previously studied a continuous deterministic model for translation, taking into account the ribosome elongation and a possible behavior similar to a “traffic jam” of ribosomes on the mRNA chain. This modeling allowed us to observe that there was a trade-off between ribosome density (fraction of an mRNA chain bound by ribosomes) and protein synthesis rate, with the optimal value in a region of elongation- and initiation-limited synthesis rate. In the present study we implemented an exact stochastic model for translation, based on a modified Gillespie’s algorithm with additional Monte-Carlo steps to follow individual stochastic behavior of all molecules. Based on this, we investigated the relation of initiation noise to the noise in protein synthesis. Additionally we examine if there is also a trade-off between optimal synthesis rate and noise. The study is performed for different ribosome densities and various mRNA lengths.