What are NBUE ( New Better Than Used in Expectation) random variables? I came across this term while reading a research paper but could not make any sense out of the information therein. Can someone please shed some light on what exactly does this mean ?
The paper that I was referring to - Models and Algorithms for Stochastic Online
Scheduling by N Megow, M Uetz, T Vredeveld (2006). In Mathematics of Operations Research 31 (3), 513-525.
On page 2 of the paper, NBUE variables are mentioned. 
URL - http://drops.dagstuhl.de/opus/volltexte/2005/110/pdf/05031.MegowNicole.ExtAbstract.110.pdf
 A: NBUE is a concept used in reliability, survival and demographic modeling. We must first have some definitions. Let $T$ be a survival time (lifetime), that is a nonnegative random variable (rv) with cdf $F$ and density $f$. Also $G(t)=1-F(t)$ is the survival function. The hazard function is 
$$
\lambda(t)=\frac{f(t)}{G(t)}
$$ with the interpretation that for small $\Delta_t$, $\lambda(t)\Delta_t$ is the conditional probability of dying in the interval $(t,t+\Delta_t]$ given survival until $t$. So $\lambda(t)$ is the rate of dying at time $t$, in demographics force of mortality. The expected life is $\DeclareMathOperator{\E}{\mathbb{E}} \mu = \E T$. Of great interest is the residual life $T_x$ which is the remaining life when reached age $x$, that is, $T_x =\max(T-x,0)$.  The expected residual life is then
$$
  \mu(x)=\E [T_x \mid T>x] = \frac{\int_x^\infty G(u)\; du}{G(x)} 
$$ 
As an example, for the exponential distribution the hazard function is constant, and the expected residual life is constant! That is related to the lack of memory property. We can say that entities (usually not biological!) with an exponentially distributed lifetime do not age, they only die by accident. 
So the exponential distribution is a natural baseline for comparison when describing life distributions. Now, finally we can answer the question. NBUE means that
$$
   \mu(0)=\mu \ge \mu(x) \quad \text{for all $x\ge 0$.}
$$
That is, mean residual life is never larger than the expected lifetime. Note that this is a rather large class of life distributions! Reversing the inequality leads to NWUE (New Worse than Used in Expectation). The exponential distribution is the sole example which is both NBUE and NWUE. 
This definitions (and much more) can be found in Chapter 2 of Maxim Finkelstein: Failure Rate Modelling for Reliability and Risk
A: A random variable X with distribution F is NBUE if
$$
\int_t^\infty (1-F(x))\;dx\,\leq \mu  (1-F(t)).
$$
where $ \mu = E X = \int_0^\infty (1-F(x)) \;dx\ $
