# Understanding Hazard Function Values Exceeding 1

I keep running into problems in understanding hazard rates. I know, for example, that in a strict sense a hazard rate is not a probability and it is continually mentioned that because of this the hazard rate has no upper bound.

Am I right in understanding that hazard function values can be interpreted as instantaneous failure rates? Given this interpretation, I still have trouble seeing how this could exceed 1.

I've seen derivations of the constant hazard rate $\lambda$ for the exponential distribution, so I see that for $\lambda \geq 1$ my question seems to have an obvious answer. But even then, is $\lambda$ not techincally a rate? I know we often just simplify and say, for instance, $\lambda = 3$. But don't we really mean $\lambda = 3$ per 1000 or 3 per 100? And if that is the case, can we really say the hazard rate is exceeding 1 in this case?

If someone could provide alternative examples where $h(t)\geq 1$ or help me understand my misconception it would be much appreciated.

• I see people's confusions on probability and probability density for survival analysis. – Deep North Nov 3 '15 at 5:53

This is perhaps best understood by looking at the exponential (time to a single event) distribution with a constant hazard rate $\lambda$, there the probability of an event by time $t$ is $1-e^{-\lambda t}$. So it is always $\in [0,1]$. In contrast the Poisson distribution is the corresponding distribution for recurring events, where the expected number of events seen in time $t$ is $t\times \lambda$.