Failure Rate Function

For a continuous positive random variable $X$ with PDF $f$ and CDF $F$ , the failure rate (or hazard rate) function is:

$r (t) = \frac{f ( t )}{1 - F ( t )}$

Interpretation

$r (t)$ represents the conditional probability density that a $t$ -year-old item will fail. Specifically:

$P (X \in (t, t + d t) ∣ X > t) \approx r (t) d t$

It answers: “Given the item has survived to age $t$ , what is the instantaneous risk of failure?”

The failure rate function uniquely determines the distribution:

$F (t) = 1 - exp (- \int_{0}^{t} r (s) d s)$

Distribution	$r (t)$	Interpretation
Exponential	$λ$ (constant)	Item doesn’t age — risk is the same at any age
Hyperexponential	Decreasing to $min λ_{i}$	As item ages, it’s more likely the “long-lived” type
Hypoexponential	Converges to $min λ_{i}$ as $t \to \infty$	Dominated by slowest component for large $t$