Non-monotonic ageing notions are looked upon as an extension of the corresponding monotonic ageing notions in this work. In particular, the New Better than Used in Expectation (NBUE) and the corresponding non-monotonic analogue New Worse then Better than Used in Expectation (NWBUE) classes of life distributions is considered. Some additional results for the NBUE class are obtained. While many properties of the NBUE class carry over in an analogous way to the NWBUE class, it is shown by means of counterexamples that the moment bounds do not. Some corrective results with respect to popular notions of the NWBUE class are also presented.