Question:
The probability \(q_x\) of dying of a person between the age interval \(x\) and \((x+1)\), and the central mortality rate \(m_x\) are related as:
The probability \(q_x\) of dying of a person between the age interval \(x\) and \((x+1)\), and the central mortality rate \(m_x\) are related as:
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Use \(L_x \approx l_x(1-q_x/2)\) under the uniform-distribution-of-deaths assumption, then invert \(m_x = d_x/L_x\) to express \(q_x\) in terms of \(m_x\).
Updated On: Jul 4, 2026
- \(q_x = 2m_x / (2 - m_x)\)
- \(q_x = 2m_x / (2 + m_x)\)
- \(q_x = m_x / (2 - m_x)\)
- \(q_x = m_x / (2 + m_x)\)
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The Correct Option is B
Solution and Explanation
Step 1: Let \(l_x\) be the number of survivors at exact age \(x\) and \(d_x = l_x - l_{x+1}\) the number of deaths in the interval \((x, x+1)\). By definition \(q_x = d_x/l_x\).
Step 2: Under the usual assumption that deaths are uniformly distributed over the year of age, a person who dies in the interval contributes on average half a year of exposure. So the person years lived in the interval (the central exposed to risk) is \(L_x = l_x - \tfrac12 d_x = l_x\left(1-\tfrac12 q_x\right)\).
Step 3: The central mortality rate is deaths divided by person years exposed: \[m_x = \frac{d_x}{L_x} = \frac{q_x l_x}{l_x\left(1-\tfrac12 q_x\right)} = \frac{q_x}{1-\tfrac12 q_x} = \frac{2q_x}{2-q_x}.\]
Step 4: Solve this for \(q_x\): \[m_x(2-q_x) = 2q_x \;\Rightarrow\; 2m_x = 2q_x + m_x q_x = q_x(2+m_x) \;\Rightarrow\; q_x = \frac{2m_x}{2+m_x}.\]
Final answer: \(q_x = 2m_x/(2+m_x)\), option (B).
Step 2: Under the usual assumption that deaths are uniformly distributed over the year of age, a person who dies in the interval contributes on average half a year of exposure. So the person years lived in the interval (the central exposed to risk) is \(L_x = l_x - \tfrac12 d_x = l_x\left(1-\tfrac12 q_x\right)\).
Step 3: The central mortality rate is deaths divided by person years exposed: \[m_x = \frac{d_x}{L_x} = \frac{q_x l_x}{l_x\left(1-\tfrac12 q_x\right)} = \frac{q_x}{1-\tfrac12 q_x} = \frac{2q_x}{2-q_x}.\]
Step 4: Solve this for \(q_x\): \[m_x(2-q_x) = 2q_x \;\Rightarrow\; 2m_x = 2q_x + m_x q_x = q_x(2+m_x) \;\Rightarrow\; q_x = \frac{2m_x}{2+m_x}.\]
Final answer: \(q_x = 2m_x/(2+m_x)\), option (B).
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