1. ## 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?

2. 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:

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

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 $\displaystyle R(t)=1-F(t)$ and that the hazard is $\displaystyle h(t)=-\frac{d}{dt}(ln(R(t)))$ that this implies the hazard of the variable is constant only if the variable is exponential?

EDIT: Nevermind, I solved it.