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We use deep Chandra X-ray imaging to measure the distribution of specificblack hole accretion rates ($L_X$ relative to the stellar mass of the galaxy)and thus trace AGN activity within star-forming and quiescent galaxies, as afunction of stellar mass (from $10^{8.5}-10^{11.5} M_\odot$) and redshift (to$z \sim 4$). We adopt near-infrared selected samples of galaxies from theCANDELS and UltraVISTA surveys, extract X-ray data for every galaxy, and use aflexible Bayesian method to combine these data and to measure the probabilitydistribution function of specific black hole accretion rates,$\lambda_{sBHAR}$. We identify a broad distribution of $\lambda_{sBHAR}$ inboth star-forming and quiescent galaxies---likely reflecting the stochasticnature of AGN fuelling---with a roughly power-law shape that rises toward lower$\lambda_{sBHAR}$, a steep cutoff at $\lambda_{sBHAR} \gtrsim 0.1-1$ (inEddington equivalent units), and a turnover or flattening at $\lambda_{sBHAR}\lesssim 10^{-3}-10^{-2}$. We find that the probability of a star-forminggalaxy hosting a moderate $\lambda_{sBHAR}$ AGN depends on stellar mass andevolves with redshift, shifting toward higher $\lambda_{sBHAR}$ at higherredshifts. This evolution is truncated at a point corresponding to theEddington limit, indicating black holes may self-regulate their growth at highredshifts when copious gas is available. The probability of a quiescent galaxyhosting an AGN is generally lower than that of a star-forming galaxy, showssigns of suppression at the highest stellar masses, and evolves strongly withredshift. The AGN duty cycle in high-redshift ($z\gtrsim2$) quiescent galaxiesthus reaches $\sim$20 per cent, comparable to the duty cycle in star-forminggalaxies of equivalent stellar mass and redshift.

X-rays across the galaxy population - II. The distribution of AGN accretion rates as a function of stellar mass and redshift