# Force of Mortality

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Does a force of mortality represent the probability that a person aged x, will die before x+h (provided h is small enough)?

Also, will a good model necessarily have an increasing force of mortality (with respect to time) since as an individual who is older is more likely to die?

answered Mar 24, 2017 by (1,490 points)
selected Mar 24, 2017 by Rowan

Hi Rohin

For a very small h, $$\mu_xh$$ can be interpreted as the probability that a newborn who has attained age $$x$$ dies between $$x$$ and $$x+h$$:

$$\mu_xh \approx Pr[x\lt X \le x+ h| X \gt x]$$

You can think of force of mortality as the conditional instantaneous measure of death at $$x$$.

With regards to your second question, a good model will not always have an increasing force of mortality. The force of mortality in fact decreases in late teenage years/early twenties and this effect is called the "accident hump". This occurs as a result of deaths from car accidents of inexperienced drivers, and drug/alcohol-related deaths. The Balducci assumption for instance exhibits decreasing force of mortality so could possibly be used in a model to explain the "accident hump". After these years, the force of mortality does increase due to individuals becoming older so more likely to die.

commented Mar 24, 2017 by (3,390 points)
Just to clarify, the decrease in the force of mortality happens after the peak of the accident hump.

commented Mar 24, 2017 by (460 points)
Thanks Josh for the first answer. Well explained.

With respect to the 2nd part:
For the "accident hump" during late teen ages, why does the force of mortality decrease and not increase (since more people die during this age period)? I thought that an increase in number of deaths in a certain period means that force of mortality will increase?

commented Mar 24, 2017 by (1,490 points)

During the "accident hump" years, the force of mortality increases up until the top of the peak of the "accident hump" but after the peak, the force of mortality decreases for a while and then increases.

commented Mar 24, 2017 by (460 points)

Okay cool. This makes sense. Thank you!!