1. The problem is to understand the concept of the force of mortality, which is a key idea in actuarial science and demography. 2. The force of mortality, denoted by $\mu_x$, is defined as the instantaneous rate of mortality at age $x$. It measures the risk of death at a very small age interval around $x$. 3. The formula for the force of mortality is: $$\mu_x = -\frac{d}{dx} \ln \ell_x$$ where $\ell_x$ is the survival function, representing the number of individuals surviving to age $x$. 4. This means $\mu_x$ is the negative derivative of the natural logarithm of the survival function. Intuitively, it tells us how quickly the survival probability is decreasing at age $x$. 5. Another way to express it is: $$\mu_x = \lim_{\Delta x \to 0} \frac{P(x \leq T < x + \Delta x \mid T \geq x)}{\Delta x}$$ where $T$ is the future lifetime random variable. This shows $\mu_x$ as the instantaneous death probability per unit time. 6. Important rules: - The force of mortality is always non-negative. - It can vary with age, often increasing as age increases. 7. To calculate or estimate $\mu_x$, you need data on survival or mortality rates at different ages. 8. Understanding $\mu_x$ helps actuaries price life insurance and pensions by modeling how likely death is at each age. Final answer: The force of mortality $\mu_x$ is the instantaneous rate of mortality at age $x$, given by $$\mu_x = -\frac{d}{dx} \ln \ell_x$$, representing the risk of death at that exact age.