Hazard rate function

Definition:

• Describe the probability of an item’s failure happen $t$, given that the item is still functioning
• Consider a positive continuous random variable $X$ be the lifetime (existing without failure) of some item.
• Let $X$ have distribution function $F$ (probability of it failing before $t$) and density $f$.
• The hazard rate function $\lambda (t)$ of $F$ is defined by $\lambda (t)=\frac{f(t)}{\bar{F}(t)}=\frac{f(t)}{1-F(t)}$
  • where $\bar{F}=1-F$ , probability of it will fail after time $t$
  • Conditional probability of (it failing in the next $dt$ time) given (not yet fail after $t$-unit old)
• The parameter λ is often referred to as the rate of the distribution.
• $F(t)$, probability of lifetime less than $t$, is: $F(t)=1-\exp (-{\int }_0^t\lambda (t)dt)$