Thermal noise measurements by space-borne antennas are commonly used to determine plasma parameters like the electron density and the plasma temperature from the noise spectra. It would be desirable to have a controlled experiment in which noise from a plasma with known properties is sampled in space and in time and which results can then be used to reproduce the satellite measurements. Here we examine the possibility to use particle-in-cell (PIC) simulations as such an experiment. In this work we present a statistically averaged noise spectrum computed with a PIC code for a simple single-Maxwellian and unmagnetized electron plasma and we compare it to both, the thermal noise spectrum for the corresponding real plasma and the noise spectrum we would anticipate from our numerical scheme. We find that we can produce noise fields with sufficiently low amplitudes to keep the plasma in a linear regime. We show that the simulation noise at low and at large wave numbers differs not only from thermal noise of a physical plasma but also from the numerical noise we would expect from our numerical scheme. We explain the drop of the noise power at low wave numbers by our initial conditions. We find experimentally the relation that connects the theoretical noise spectrum for our simulation code with that we actually measure, provided that the phase velocity of the noise is less than the maximum velocity of the computational particles.