Many regulatory molecules are present in low copy numbers per cell so that significant random fluctuations emerge spontaneously. Because cell viability depends on precise regulation of key events, such signal noise has been thought to impose a threat that cells must carefully eliminate. However, the precision of control is also greatly affected by the regulatory mechanisms' capacity for sensitivity amplification. Here we show that even if signal noise reduces the capacity for sensitivity amplification of threshold mechanisms, the effect on realistic regulatory kinetics can be the opposite: stochastic focusing (SF). SF particularly exploits tails of probability distributions and can be formulated as conventional multistep sensitivity amplification where signal noise provides the degrees of freedom. When signal fluctuations are sufficiently rapid, effects of time correlations in signal-dependent rates are negligible and SF works just like conventional sensitivity amplification. This means that, quite counterintuitively, signal noise can reduce the uncertainty in regulated processes. SF is exemplified by standard hyperbolic inhibition, and all probability distributions for signal noise are first derived from underlying chemical master equations. The negative binomial is suggested as a paradigmatic distribution for intracellular kinetics, applicable to stochastic gene expression as well as simple systems with Michaelis-Menten degradation or positive feedback. SF resembles stochastic resonance in that noise facilitates signal detection in nonlinear systems, but stochastic resonance is related to how noise in threshold systems allows for detection of subthreshold signals and SF describes how fluctuations can make a gradual response mechanism work more like a threshold mechanism.