Table-top laser-driven hard x-ray sources with kilohertz repetition rates are an attractive alternative to large-scale accelerator-based systems and have found widespread applications in x-ray studies of ultrafast structural dynamics. Hard x-ray pulses of 100 fs duration have been generated at the Cu K α wavelength with a photon flux of up to 109 photons per pulse into the full solid angle, perfectly synchronized to the sub-100-fs optical pulses from the driving laser system. Based on spontaneous x-ray emission, such sources display a particular noise behavior which impacts the sensitivity of x-ray diffraction experiments. We present a detailed analysis of the photon statistics and temporal fluctuations of the x-ray flux, together with experimental strategies to optimize the sensitivity of optical pump/x-ray probe experiments. We demonstrate measurements close to the shot-noise limit of the x-ray source.