Root locus analysis is a graphical method to determine how the roots of the characteristic equation of a linear time-invariant feedback loop change with the loop gain. In this paper we show that a similar analysis can be carried out for randomly sampled systems, i.e., controlled linear systems sampled at random times spaced by independent and identically distributed time-varying intervals. For such systems, the roots of a characteristic equation determine the behavior of expected values of signals in the loop. The root locus analysis in this context is especially useful for positive systems, for which (almost sure) stability can be concluded if the roots of the characteristic equation have a negative real part, and it is particularly simple when the distribution of the intervals between sampling times is exponential or Erlang.