# Relation between Hazard rate and expected time until next failure.

For an exponential distribution, the hazard rate has a clear interpretation as the inverse of the expected time until the next failure (let's say we are modelling machine failures here). For other distributions however, this does not seem to hold. Take for example, the Lomax distribution with probability density function -

$$f(x) = \frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}$$

The Hazard rate becomes -

$$H(x) = \frac{\lambda \kappa}{1+\lambda x}$$

And the expected value of the random variable being modelled given it is greater than $x$ is -

$$E[X|X>x] = \frac{1}{\lambda (\kappa-1)}+ \frac{\kappa x}{\kappa-1}$$

Unlike the exponential distribution,

$$H(x) \neq \frac{1}{E[X|X>x] - x}$$ (see comment from @Math1000 below)

Although they are both linear.

So, what is the difference between these two quantities and why are they the same only for the exponential distribution?

• Recall that the exponential distribution is the unique continuous probability distribution with the memoryless property. So if $T\sim\mathrm{Exp}(\mu)$ then for $t>0$ ,$$\mathbb E[T\mid T>t] = \mathbb E[T] + t = \frac1\mu + t.$$ Commented Feb 20, 2017 at 7:08
• What I'm wondering is, what do I make of the hazard rate (in general) if not the inverse of the expected time until the next failure? Commented Feb 20, 2017 at 7:13
• Updated my post with the correction you pointed out. Commented Feb 20, 2017 at 7:17
• The hazard function $h(t)$ is the instantaneous rate of occurrence of failure at time $t$. The exponential distribution is has a constant hazard rate (which is the reciprocal of its mean) precisely because of its unique memoryless property. Commented Feb 20, 2017 at 7:23
• What I'm struggling with is - if h(t) is the instantaneous rate of occurrence of failure, then shouldn't $1/h(t)$ be instantaneous expected time till the next failure? Or is that completely off? Commented Feb 20, 2017 at 7:29

The instantaneous hazard rate and additional expected time are the same only for cases of deterministic processes or the exponential distribution.

Otherwise, the hazard rate is like instantaneous velocity and the reciprocal of additional expected time is like average velocity.