Residence times of conservative pollutants in a lake or a reservoir are studied without the usual approximation of constant inflow/outflow. The effects of a random discharge flushing out a reservoir are investigated with the techniques of stochastic differential equations. Stochastic properties of the concentration in the reservoir are derived from stochastic properties of discharge (moments, autocorrelation function, and probability distribution function), and some approximations are analyzed. The major results are two simple and usable corrections for the residence time in two limiting cases: when autocorrelation time of discharge is much shorter than reservoir residence time and, at the opposite, when it is much longer.