Why is density function of survival time called instantaneous failure rate

Question

Why density function of survival time is called instantaneous failure rate. I am confused why density function is a failure rate for survival time.

Answer 1

According to Wikipedia:

Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time.

If all individuals eventually fail and there can be only 1 failure per individual, then this is simply a 1-dimensional probability distribution. The proportion that has failed at any one time is the (cumulative) probability distribution of failure evaluated at that time, $F(t)$; the surviving proportion is 1 minus that, $S(t) = 1-F(t)$. In continuous time, the instantaneous failure rate is the density function, $f(t)=dF(t)/dt$, as in the comment by @mlofton. Thus the terminology "survival function" can be applied to any 1-dimensional probability distribution, even outside situations that might usually be considered "survival analysis."

What's often confusing is that this instantaneous failure rate is not the same thing as what is usually meant by the seemingly similar word "hazard." That's usually meant to be the instantaneous failure rate for those who have already survived that long, $h(t)=f(t)/S(t)$. If "hazard" is what you had in mind, then your confusion is quite understandable.