Interestingly, life expectancy results as expressed by the forward probability distribution function, the one which, under simplifications was used since the 1930s of the las century to express, in hydrology, the instantaneous unit hydrograph (IUH). In this case, however, the IUH was usually taken as time-invariant, and (11) represents its generalisation. For those who miss it, \(R\) is know just if we make strong assumptions about the future. It is clear in fact, that we should know all the inputs still to come to get constructively the forward probability. Otherwise, the approach followed in the past was to assume a form of \(p_{ex}\) directly, as chosen from one of the known distribution function families, and verify a-posteriore its fitting to data.
S(t,t')
The function \(q\left(t_{in},t_{ex}\right)\), integrated over \(t_{in}\), it gives all the discharge at \(t_{ex}\):