Half-life estimation under indefinite 'mother-daughter' relation

An algorithm has been proposed to build an estimate of the half-life of a “daughter” nucleus in case, when it is unknown, which nucleus is its “mother” (“indefinite start time”). For a decay of the “mother” at an instant t we can determine P—a probability of such a decay, if we assume that each “mother”, which has been decayed before t has equal chances to be “mother” of this “daughter”: $$ \sum\limits_{i = 1}^{n_t } {exp\left( { - \frac{{t - t_i }} {\tau }} \right)} , $$ where τ is the life-time of the “daughter” and nt is their total number; ti ist the instant of the “mother” decay, such that ti ⩽ t.