The formula for the central rate of mortality is given as :

and the formula for the force of mortality is given as :

Are these not equivalent?

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Hi

So you actually ask an interesting question, and what it comes down to is the difference between what something represents does not necessarily imply a difference in their estimators.

So the force of mortality, \(\mu_x \), represents the instantaneous rate of transition from alive to dead (i.e. the probability of dying right now given you are alive), and is estimated from data as \(\frac{d_x}{E^c_x}\).

On the other hand, the central rate of mortality, \(m_x \), represents the "average deaths" per person aged x (look back at the slides in the first section). It is calculated by \(\frac{number of deaths}{time spent alive and at risk} \).

Thus the estimates of these two happen to be the same, yet they represent completely different concepts. Furthermore, the rate of mortality is a constant for an age x, whereas the true force of mortality is usually a function of t for an age x.

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