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Given a reliability function $$R(t) = e^{-(\alpha)t},$$

what would be its failure time distribution?

I am not sure if failure time distribution is the correct wording for this either.

Is it the hazard function $h(t) = f(t)/R(t)$ ?

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If this is homework, please add the homework tag. $R(t)$ is the probability that the system survives up to time $T$. Thus, if $L$ is the time at which the system fails, $P\{L > t\} = R(t)$. You can get the distribution of the failure time $L$ from this. –  Dilip Sarwate Feb 8 '12 at 2:36
Hello Dilip, no this is not homework. This is just a question in order to solidify my understanding with Reliability functions, hazard functions and failure time distributions. –  Alistair Feb 8 '12 at 2:45
OK, so you know $P\{L > t\} = 1 - F_L(t) = R(t)$. Can you get $f_L(t) = \frac{dF_L(t)}{dt}$ from that? –  Dilip Sarwate Feb 8 '12 at 2:54
Wouldn't the failure distribution be just 1 - FL(t) ? Do we need to get the derivation of FL(t)? –  Alistair Feb 8 '12 at 3:30
If it can be done, it is good to find the density function, since it is useful in many calculations. And yes, to find the density function $f_L(t)$, you differentiate the cumulative distribution function $F_L(t)$ with respect to $t$. –  André Nicolas Feb 8 '12 at 6:26

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