Compound Poisson With Erlang Damage

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

j=o and

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

Thus, and

Thus

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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