How to find Reliability of a rectangular distribution function?

Question

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is:

f(x) = {1/b-a if a<= t <= b

       0  otherwise}

Can somebody help me out in finding the Reliability function R(t) for that device.

Thanks

Answer 1

Just perform the computation using: $R(t) = p \{\omega | T(\omega) > t \} = \int_t^\infty f(x) dx$.

It may help to draw $f$ and think about what $R(t)$ means in terms of the graph of $f$.

It should be clear that no devices fail for $t<a$, and that all devices fail for $t>b$. This tells you what $R(t)$ is for these ranges.

Remember that $f$ is a pdf.

Consider three cases:

(1) $t<a$:

We have $R(t) = \int_t^\infty f(x) dx = \int_a^b f(x) dx = 1$.

(2) $t \in [a,b]$:

$R(t) = \int_t^b f(x) dx = {b -t \over b-a}$.

(3) $t >b$:

$R(t) = \int_t^\infty f(x) dx = 0$.