# BUS3018F: reconciling the mathematical specification of the initial exposed to risk with its definition

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Hi all!

Let's get a short discussion going on the initial exposed to risk. Recall that $$E_{x}$$ arises as a mathematical construct in the generalized binomial model (you don't have to worry about the derivation), and so the mathematical definition is "the summation of the central exposed to risk (which is a period of time) and (1-$$t_{i}$$)- which is also a period of time - for all the deaths".

On the other hand, the qualitative definition of $$E_{x}$$ is "the equivalent (or I would say expected) number of lives which would have to be alive at the beginning of the rate interval (e.g. exact age $$x$$)  in order to have given rise to the number of deaths observed, had there been no entrants or exits (other than death).

Now, how can we reconcile the mathematical definition of $$E_{x}$$ with the qualitative definition? Also, how can $$E_{x}$$ approximate a number of lives when it is only a summation of time?

I am interested to see YOUR responses (including tutors or others not involved in the course) please. Give it a bash!