Detailed description of a two-state non-Markov system

A two-state system in which transitions between the states are made randomly, is often used as a model of different phenomena in several fields. Both the probability density for the cumulative dwell-time in one of the states during a fixed observation time T, and the distribution of the number of transitions made during that time are known. We calculate the joint distribution of these two correlated random variables, showing that at long times it approaches a two-dimensional Gaussian form.