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In-Depth Information
Pr
(
A
∪
B
)
=
Pr
(
A
|
B
)
Pr
(
B
).
(3.216)
Now, if we identify
A
with a failure in the specified interval and
B
with no failure up to
time
t
,wehave
Pr
(
A
∪
B
)
=
Pr
(
t
<τ
≤
t
+
t
)
=
ψ(τ)
t
.
(3.217)
Here
ψ (τ)
is the probability distribution density for the first failure to occur in the time
interval
(
t
,
t
+
t
)
. We also introduce the survival probability
(
t
)
through the integral
∞
t
ψ(τ)
(
)
=
τ,
t
d
(3.218)
so that the probability that a failure has not occurred up to time
t
is given by
Pr
(
B
)
=
Pr
(
t
<τ)
=
(
t
).
(3.219)
Consequently the age-specific failure rate (
3.215
) is given by the ratio
)
=
ψ(
t
)
g
(
t
)
.
(3.220)
(
t
As a consequence of the integral relation (
3.218
) we can also write
d
(
)
dt
,
t
ψ(
t
)
=−
(3.221)
which, when inserted into (
3.220
), yields the equation for the rate
d
ln
[
(
)
]
dt
,
t
g
(
t
)
=−
which is integrated to yield the exponential survival probability
exp
t
(
)
=
−
(τ)
τ
.
t
g
d
(3.222)
0
Note that the probability density
ψ(
t
)
is properly normalized,
∞
0
ψ(τ)
τ
=
,
d
1
(3.223)
since it is certain that a failure will occur somewhere between the time extremes of
zero and infinity. The normalization condition is consistent with the survival probability
having the value
(
=
)
=
;
t
0
1
(3.224)
that is, no failures occur at time
t
=
0.
