# Exposed to risk: Past test question

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Past test question(2014 Q4 g):

"Suppose that E_x = E_x^c  + \frac{3}{4}  d_x. Give, with assumptions, the age to which each of the estimates of the probability of death and the force of mortality for lives classified x would apply. "

The answer is that q_x is estimated by \frac{d_x}{E_x} and mu_x+\frac{1}{4} is estimated by \frac{d_x}{E_x^c}. Please explain how these answers were concluded.

answered Jun 19 by (2,610 points)

Hi

So I think you need to really consider what the initial exposed to risk represents. Initial exposed to risk often includes the deaths that are expected to occur in the first half of the year. Therefore if we are including 0.75 of deaths, that means we occur 75% of deaths in the first half of the year. Thus on average, deaths are expected to occur 0.25 into the year (i.e half of the deaths occur in the first three months).

Now using this, consider the average age of death and the average age at the beginning of the interval, to find the value of the x that qx estimates. I can give you the answer, but I would rather you find it yourself.