# Failure Rate Function

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

## References

### 2013

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

The failure distribution function is the integral of the failure density function, f(t), :$F(t)=\int_{0}^{t} f(\tau)\, d\tau. \!$ The hazard function can be defined now as :$h(t)=\frac{f(t)}{R(t)}. \!$ 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, :$F(t)=\int_{0}^{t} \lambda e^{-\lambda \tau}\, d\tau = 1 - e^{-\lambda t}, \!$ which is based on the exponential density function. The hazard rate function for this is: :$h(t) = \frac{f(t)}{R(t)} = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda .$

### 1963

• (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 $F$ (or its density $f$) to properties of the corresponding hazard rate $q$ defined for $F(x)\lt 1$ by $q(x)=f(x)/[1−F(x)]$. It is shown, e.g., that the class of distributions for which $q$ is increasing is closed under convolution, and the class of distributions for which $q$ is decreasing is closed under convex combinations. Using the fact that $q$ is increasing if and only if $1-F$ is a Polya frequency function of order two, inequalities for the moments of $F$ are obtained, and some consequences of monotone $q$ for renewal processes are given. Finally, the finiteness of moments and moment generating function is related to limiting properties of $q$.