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In the life table, two values are related to death. One is the death rate, by definition,

$$ m_x = \frac{d_x}{n_x}$$

Another one is the probability of dying:

$$ q_x = \frac{d_x}{l_x}$$

The numerator of these two formulas are the same, but the denominators are different.

I cannot understand the difference between $l_x$ and $n_x$ even though I read the notes many times...

Can someone explain it to me in a more straightforward way?

User1865345
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doraemon
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1 Answers1

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$l_x$ is the number of persons out of the cohort living at the specified age $x.$

$d_x$ is the number of persons out of $l_x$ who die before attaining age $x+1, $ i.e. $d_x=l_x-l_{x+1}.$

So, $q_x$ measures the the probability of a person of exact age $x$ dying within one year.

$m_x$ in a life table measures the probability that a person whose exact age is not known but lying in $(x, x+1) $ would die within one year. The denominator is actually (provided the deaths are uniform) $L_x:= \int_0^1 (l_x -td_x) ~\mathrm dt=l_x-\frac12d_x.$

In case of stationary population, number of persons in the age group $(x, x+1) $ would be generally denoted by $n_x.$

User1865345
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  • Can I understand in such a way: $q_x$ is a probability that conditions on exact age x? – doraemon Mar 09 '23 at 06:16
  • Actually, I am studying abridged life table, which was created by Chiang. I am so confused that whether the notation of the abridged life table is the same as that I mentioned in the question... – doraemon Mar 09 '23 at 06:21
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    Yes @doraemon. That could be a legit interpretation. – User1865345 Mar 09 '23 at 07:02