Analytical formulations of critical droughts either as longest drought duration or maximum deficit sum have been derived by making use of the theories of random number of random variables and runs. The results obtained are valid for independent and lag-one Markov processes only. The numerical solution of derived analytical expressions yields any order of statistical moments the most important one is the expected value of critical droughts. It has been quantitatively observed that the drought duration and severity are dependent on the correlation structure, sample size, and truncation level.