I am stuck on the following problem:
(a) Construct a table of $Px$ for Makeham's law with parameters $A= 0.0001, B = 0.00035$ and $C = 1.075,$ for integer $x$ from age $0$ to age $130,$ using Excel or other appropriate computer software. You should set the parameters so that they can be easily changed, and you should keep the table, as many exercises and examples in future chapters will use Makeham's law.
(b) Use the table to determine the age last birthday at which a life currently aged $70$ is most likely to die.
I have found the solutions manual that solves part (b) in the most "descriptive" way to leave me still completely confused. I can't take anything from this to be able to use this to solve similar problems. Please help!
"The probability that $(70)$ dies at age $70 + k$ last birthday is $Pr[K70 = k]$ where $Kx$ is the curtate future lifetime. The most likely age at death is the value of $k$ that maximizes $Pr[K70 = k] = k\mid q70.$
The maximum value for $k \mid q70$ can be found by constructing a table of values and selecting the largest value; it is $3 \mid q70 = 0.05719,$ so the most likely age at death is $73$"
I still have no idea how that was deemed as the largest value.
Unfortunately, you did not include the explicit parametrization for Makeham's law as used in the text. As a result, I had to locate a copy of the full exercise. In the future, if you want to maximize your chance of having your question answered, you should include all relevant information from the question, not just select portions.
Makeham's law as described in the exercise is $$\mu_x = A + Bc^x, \tag{1}$$ where $\mu_x$ is the hazard rate or force of mortality for a life aged $x$. The exercise then gives the formula $${}_t p_x = e^{-At} (\exp(-B/\log c))^{c^x (c^t - 1)} \tag{2}$$ for the $t$-year survival probability of a life aged $x$. This allows us to produce a survival table for $(70)$ for the choices $x = 70$, $A = .0001$, $B = .00035$, $c = 1.075$: $$\begin{array}{c|c|c|c} t & {}_t p_{70} & {}_t q_{70} = 1 - {}_t p_{70} & {}_{t|}q_{70} = {}_{t+1} q_{70} - {}_t q_{70} \\ \hline 1 & 0.944178 & 0.055822 & 0.565313 \\ 2 & 0.887647 & 0.112353 & 0.0569955 \\ 3 & 0.830652 & 0.169348 & \color{red}{0.0571894} \\ 4 & 0.773462 & 0.226538 & 0.0570894 \\ 5 & 0.716373 & 0.283627 & 0.0566753 \\ 6 & 0.659698 & 0.340302 & 0.0559306 \\ 7 & 0.603767 & 0.396233 & 0.054844 \\ 8 & 0.548923 & 0.451077 & 0.0534102 \\ 9 & 0.495513 & 0.504487 & 0.051631 \\ 10 & 0.443882 & 0.556118 & 0.0495162 \\ \vdots & \vdots & \vdots & \vdots \end{array}$$ Thus for $(70)$, the most likely curtate age of death corresponds to the value of $x+t$ for which ${}_{t|}q_{70}$ is largest.