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:

• A. \(q_x = 2m_x / (2 - m_x)\)
• B. \(q_x = 2m_x / (2 + m_x)\)
• C. \(q_x = m_x / (2 - m_x)\)
• D. \(q_x = m_x / (2 + m_x)\)

Hint:

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 \(m_x\).

Answer 1

The Correct Option is B

Step 1: It is a standard actuarial identity that the reciprocal of the exposure per death splits into the reciprocal of \(q_x\) minus half a year, since exposure per death equals the average years lived by all \(l_x\) persons divided by deaths, adjusted for those who die mid year: $\dfrac{1}{m_x} = \dfrac{1}{q_x} - \dfrac12$.
Step 2: Rearranging this identity gives $\dfrac{1}{m_x} = \dfrac{2-q_x}{2q_x}$, which on inverting reproduces $m_x = \dfrac{2q_x}{2-q_x}$, consistent with the definition of the central rate.
Step 3: Now solve directly for $q_x$ from $\dfrac{1}{q_x} = \dfrac{1}{m_x}+\dfrac12 = \dfrac{2+m_x}{2m_x}$.
Step 4: Inverting both sides gives $q_x = \dfrac{2m_x}{2+m_x}$, matching the result obtained from the exposure based derivation.
\[ \boxed{q_x = \dfrac{2m_x}{2+m_x}} \]