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Emission models that form the basis of self-consistent field computations make use of the approximation that emitted electrons form a smooth space charge jelly. In reality, electrons are discrete particles that are subject to statistical Coulomb interactions. A Monte Carlo simulation tool is used to evaluate the influence of discrete space charge effects on self-consistent calculations of cathode-ray tube optics. We find that interactions in the space charge cloud affect the electron trajectories such that the velocity distribution is Maxwellian, regardless of the current density. Interactions near the emitter effectively conserve the Maxwellian distribution. The surprising result is that the width of the distribution of transversal velocities does not change. The distribution of longitudinal velocities does broaden, as expected from existing theories.