Failure Rate Function

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A Failure Rate Function, [math]F(t)[/math], is a temporal probability function that produces a failure probability (the probability that a case will fail in time interval [math]t[/math]).



    • Calculating the failure rate for ever smaller intervals of time, results in the Template:Visible anchor (loosely and incorrectly called hazard rate), [math]h(t)[/math]. This becomes the instantaneous failure rate as [math]\scriptstyle\Delta t [/math] tends to zero: :[math]h(t)=\lim_{\triangle t \to 0} \frac{R(t)-R(t+\triangle t)}{\triangle t \cdot R(t)}.[/math] A continuous failure rate depends on the existence of a failure distribution, [math]\scriptstyle F(t)[/math], which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t, :[math]\operatorname{Pr}(T\le t)=F(t)=1-R(t),\quad t\ge 0. \![/math] where [math]{T}[/math] is the failure time.

      The failure distribution function is the integral of the failure density function, f(t), :[math]F(t)=\int_{0}^{t} f(\tau)\, d\tau. \![/math] The hazard function can be defined now as :[math]h(t)=\frac{f(t)}{R(t)}. \![/math] Many probability distributions can be used to model the failure distribution (see List of important probability distributions). A common model is the exponential failure distribution, :[math]F(t)=\int_{0}^{t} \lambda e^{-\lambda \tau}\, d\tau = 1 - e^{-\lambda t}, \![/math] which is based on the exponential density function. The hazard rate function for this is: :[math]h(t) = \frac{f(t)}{R(t)} = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda .[/math]





  • (Barlow et al., 1963) ⇒ Richard E. Barlow, Albert W. Marshall, and Frank Proschan. (1963. "Properties of probability distributions with monotone hazard rate.” In: The Annals of Mathematical Statistics, 34(2).
    • ABSTRACT: In this paper, we relate properties of a distribution function [math]F[/math] (or its density [math]f[/math]) to properties of the corresponding hazard rate [math]q[/math] defined for [math]F(x)\lt 1[/math] by [math]q(x)=f(x)/[1−F(x)][/math]. It is shown, e.g., that the class of distributions for which [math]q[/math] is increasing is closed under convolution, and the class of distributions for which [math]q[/math] is decreasing is closed under convex combinations. Using the fact that [math]q[/math] is increasing if and only if [math]1-F[/math] is a Polya frequency function of order two, inequalities for the moments of [math]F[/math] are obtained, and some consequences of monotone [math]q[/math] for renewal processes are given. Finally, the finiteness of moments and moment generating function is related to limiting properties of [math]q[/math].