Statistics for Maintenance Modelling: Renewal Processes

# Proving the Identity of M(t) More Intuitively¤

The expected number of failures in an interval dt is v(t)dt. Since this number must be 0 or 1, v(t)dt is also the probability that a failure occurs in dt. Then the probability that one does not occur is 1 - v(t)dt and the expected number is 1 - v(t)dt 0 + v(t)dt 1 = v(t)dt.

The probability that the first failure occurs in dt is f1(t) dt and so on, so that the probability that the 1st, or 2nd, or 3rd, or 4th, or indeed that any failure occurs in dt is fk(t) dt.

For the exponential example, without any integrating by parts, we had fk(t), and so:

v(t) = (t)k/k! = , (since (t)k/k! = )
whence M(t) = du = t.

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