I am currently going through some actuarial lecture notes which, without definition, use the term 'central exposed to risk', which it denotes by $E_{x,t}^c$. Having googled this term, the most easily understandable explanation of this term seems to imply that if we have a population of initial size $p$ at time $x$ and at time $x+t$ the size of this population is $q$ then $$ E_{x,t}^c = p - \frac{p-q}{t} $$ i.e. the average number of lives which are alive within the population between times $x$ and $x+t$.
Is this correct?
Given a life with age label $x$, its central exposed for risk is the waiting time from date $x$ until that life exits.
The ‘central exposed to risk’ for a population $p$ in the time interval $[x,t]$ is the sum of the individual central exposed to risk.
The formula above estimates the central exposed to risk for a population $p$ assuming that the distribution of exits ($p-q$) is uniform within the period.