Suppose now that the amount of damage in each occurrence is a random variable having an Erlang (p, k) distribution, k > 2. In this case the reliability function is to
R(t; A, Z, k) = £ Pj; At)(1 - P (jk - 1; Z)), (30)
j=o where Z = P0. Changing the order of summation yields
where [a] is the maximal integer not exceeding a. The corresponding pdf of T(0) is
Define the probability weights
Notice that Aj (k, Z) > 0 for all j = 0,1,... and ^^Aj(k; Z) = 1. One can j=o write
The hazard function is the ratio of (35) over (34). As in Zacks [Z04] one can prove that lim h(t; A,Z,k) = A
The moments of T(0) are given by
Moreover, the increase of h(t; X, Z, k) from XP(k — 1; Z) to A is monotone.
In the double stochastic case, where A is distributed like Gamma(A*,^), the reliability function and pT(t; A*,Z,k,v) is as in (24) and (26) in which p(j; Z) is replaced by Aj(k; Z).
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