Prescriptions for the stochasticity effect on the integrated X-ray luminosity of star-forming galaxies:Implications for selecting star-forming galaxies and AGN in X-ray surveys

(abridged) The integrated X-ray luminosity (Lx) of star-forming galaxies is dominated by high-mass X-ray binary (HMXB) populations. The discrete nature of these populations introduces stochastic sampling effects that distort the X-ray Luminosity Function (XLF) and bias observed scaling relations. We investigate how stochastic sampling of the HMXB XLF affects the predicted integrated Lx across a wide range of star-formation and metallicity conditions, quantifying the scatter to provide a statistical framework for interpreting X-ray observations. Using Monte Carlo simulations, we derive Lx distributions over a broad grid of star-formation rate (SFR) and metallicity values. By measuring statistical quantities describing these distributions, we parametrize the luminosity scatter by fitting surfaces to the upper and lower Lx bounds as functions of SFR and metallicity. We provide practical prescriptions to compute the expected Lx for given SFR and metallicity, fully accounting for stochastic effects without rerunning costly XLF sampling. Applying these to local and high-redshift samples shows stochasticity must be considered before attributing Lx differences to intrinsic properties. A simulation study across z=0.5-5 reveals mild redshift evolution of stochastic scatter, with minimum scatter at z 2.5. Our prescriptions quantify biases in scaling relations introduced by flux-limited surveys. At low redshifts, stochastic effects can raise Lx by up to 1 dex, overlapping with the low-luminosity AGN regime and biasing source classification in deep surveys. These prescriptions offer a framework for constraining scatter, quantifying extreme outliers, and refining X-ray source classification in current and future surveys.