Expected failure time for an item
whose maintenance policy is time-based
Let 
be the time of a
cycle, 
the preventive
maintenance time interval, and 
the
failure time.
The expected life cycle will
be the planned maintenance time multiplied by the
probability that planned maintenance does occur, plus the expected failure time
(knowing that failure occurs before tp) multiplied by the
probability that failure occurs before tp.
This is expressed mathematically by:
Equation 1
The term,
in Equation 1, is the expected time to
failure, given that failure occurs prior to scheduled maintenance, under a
policy where scheduled maintenance is carried out at time .
We wish to show that it can be expressed as
First, we recognize that the conditional distribution
function of 
is
Equation 2
In the first part of Equation 2, we have simply defined the
distribution function of (T≤t
given that T≤tp) as 
. We will call this
conditional distribution function, “Fc(t)”. (Recall that his is the
definition of a distribution function)
Now, moving towards the right in Equation 2, the top condition “1, where
t>tp” is easy to understand. We know that failure will have
occurred prior to tp (with 100% certainty) because T≤tp is our hypothesis in Fc(t).
The bottom condition 
requires
us to know that the conditional probability P(A|B) is 
where
A = T≤t and B = T≤tp
But we know that the intersection of T≤t and T≤tp
is T≤t (see footnote[1])
In the rightmost part of Equation 2 we apply the definition of F(t) to the numerator and
denominator. And, of course, we know that F(t) = 1-R(t).
Then the conditional density function of is
Equation 3
We have used, in Equation 3, the fact that the density function is the first
derivative of the distribution function.
Therefore,
Equation 4
Here, in
Equation 4, we have invoked the definition of “Expectation” as the integral of
the product of t and the density function.
[1] Because t≤tp, T≤t is entirely within the probability
space of T≤tp. Therefore
the intersection of T≤t and T≤tp
is actually T≤min(t,tp)=T≤t.