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?

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.
