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Math Help - Hazard Function Question

  1. #1
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    Hazard Function Question

    I am trying to show that the hazard function h(t)=-d/dt(ln(R(t))) is constant if and only if the variable has an exponential distribution where R(t) is the reliability 1-F(t) of a random variable with cdf F. I am not totally sure where to start in order to prove this in either direction.

    Is it right to simplify to h(t)=F'(t)/(1-F(t)) and plug in?
    Last edited by jass10816; February 1st 2010 at 07:28 PM.
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  2. #2
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    Okay, so given that it is an exponential distribution, I can prove that the hazard is constant, since I know the cdf and pdf of an exponential:

     -\frac{R'(t)}{R(t)}=\frac{-R(t)}{1-(1-exp(-xt))}=\frac{F'(t)}{exp(-xt)}=\frac{x*exp(-xt)}{exp(-xt)}=x

    However, can someone help with how to prove in the other direction. i.e. how to show that given that the reliability of a random variable with cdf F is defined to be R(t)=1-F(t) and that the hazard is h(t)=-\frac{d}{dt}(ln(R(t))) that this implies the hazard of the variable is constant only if the variable is exponential?


    EDIT: Nevermind, I solved it.
    Last edited by jass10816; February 3rd 2010 at 07:14 AM.
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