TITLE:
A Finite Failure Rate
AUTHORS:
Jaskeerat Singh
KEYWORDS:
Statistical Distribution, Truncations, Inverse Power Law, Stochastic, Waiting Time, Failure Rate, Renewal Events
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
26,
2026
ABSTRACT: We study a stochastic truncated inverse power-law waiting-time distribution, originally motivated by bounded note durations in musical structure. The model generates waiting times
τ∈[
T
min
,
T
max
]
from a uniform random variable and produces a normalized inverse power-law probability density with exponent
μ>1
. For this bounded renewal process, we derive expressions for the Cumulative Distribution Function, the Probability Density Function, the Survival Probability, and the corresponding Failure Rate (hazard function) in the sense of Cox. The failure rate is finite at
τ=
T
min
but diverges as
τ→
T
max
, mathematically enforcing the certainty that an event occurs before the maximal time scale. We also obtain the first moment explicitly, including the
μ→2
limit via L’H?pital’s rule, and emphasize that all moments are finite due to truncation. These results clarify the structure of bounded heavy-tailed renewal statistics and provide a framework for applications in complex systems where hard lower and upper time-scale constraints are present.