Mean residual life in terms of incomplete moments In a paper by Braga, Cordeiro, and Ortega 2017, they representing the Mean Residual Life in Terms of the incomplete moments
$$\operatorname{MRL}(t)= \frac{1-m_1(t)}{1-F(t)}-t$$
where $m_1(t)=\int_0^t x f(x) \, dx$
How can we get this result?
 A: Suppose $f(x)\,dx,$ for $x\ge0,$ is the probability distribution of a random variable denoted by (capital) $X$ and $F$ is the cumulative probability distribution function.
If $A\subseteq [t,+\infty)$ then 
$$
\Pr(X\in A\mid X>t) = \frac{\Pr(X\in A\ \&\ X>t)}{\Pr(X>t)} = \frac{\Pr(X\in A)}{\Pr(X>t)} = \frac{\int_A f(x)\,dx}{1-F(t)}.
$$
Therefore the conditional probability distribution of $X$ given that $X>t$ is 
$$
\frac{f(x)\, dx}{1-F(t)} \quad \text{for } x> t.
$$
Consequently we have the conditional expected value
$$
\operatorname E(X\mid X>t) = \frac{\int_t^\infty xf(x)\,dx}{1-F(t)}.
$$
This is readily seen to be the same as
$$
\frac{\int_0^\infty xf(x)\,dx - \int_0^t xf(x)\,dx}{1-F(t)}. \tag 1
$$
If we have
$$
\operatorname E(X) = \int_0^\infty xf(x)\,dx = 1
$$
then line $(1)$ above is the same as
$$
\frac{1-m_1(t)}{1-F(t)}.
$$
(And subtraction of $t$ is done because the residual lifetime is the time after $t.$)
Did this appear in some context in which the expected lifetime would be $1\text{?}$
