How do we prove that the failure rate function $\displaystyle r(t) = \frac{f(t)}{1-F(t)} $ uniquely determines the distribution $\displaystyle F $? I wrote $\displaystyle r(t) = \frac{\frac{d}{dt} F(t)}{1-f(t)} $.

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- Jan 28th 2008, 12:46 PMshilz222Failure Rate Function-Stats
How do we prove that the failure rate function $\displaystyle r(t) = \frac{f(t)}{1-F(t)} $ uniquely determines the distribution $\displaystyle F $? I wrote $\displaystyle r(t) = \frac{\frac{d}{dt} F(t)}{1-f(t)} $.

- Jan 28th 2008, 01:52 PMTKHunny
?? Why did the denominator change?

Can't you just solve for F?

Consider R(t) = ln(1-F(t)) with r(t) = R'(t). - Jan 28th 2008, 08:33 PMshilz222
typo