Exercise 1. Suppose that (T(x)) is uniformly distributed (DeMoivre) over ((0,w_x-x)), (T(y)) is uniformly distributed over ((0,w_y-y)), and that (T(x)) and (T(y)) are independent. Let (w_x=100), (x=30), (w_y=110), and (y=28). Determine:
begin{eqnarray*}
~_t p_x = 1 -frac{t}{w_x-x}, ~_t p_x mu_{x+t} = frac{1}{w_x-x}, ~textrm{and}~ mu_{x+t} = frac{1}{w_x-x-t}.
end{eqnarray*}
Thus,
begin{eqnarray*}
~_{20} p_{30:28} &=& ~_{20} p_{30}^m ~_{20} p_{28}^f = left(1-frac{20}{100-30}right)
left(1-frac{20}{110-28}right) = 0.587.
end{eqnarray*}
a) (~_{20} p_{30:28})
begin{eqnarray*}
mu_{30:28}(20) = mu_{30+20}^m + mu_{28+20}^f = frac{1}{100-50}+frac{1}{110-48} = 0.0306. end{eqnarray*}
b) ( mu_{30:28}(20))
begin{eqnarray*}
~_{20} p_{overline{30:28}} & =&
~_{20} p_{30}^m + ~_{20} p_{28}^f – ~_{20} p_{30:28}
& =& frac{50}{70} + frac{62}{82} – 0.587 = 0.9457
end{eqnarray*}
c) (~_{20} p_{overline{30:28}})
Exercise 2. Suppose that (T(x)) is exponentially distributed with force of mortality (mu_x(t)=mu_1=0.02), (T(y)) is exponentially distributed with force of mortality (mu_y(t)=mu_2=0.015), and that (T(x)) and (T(y)) are independent.
Provide expressions for:
begin{eqnarray*}
~_t p_x = exp(-mu_1 t).
end{eqnarray*}
Thus,
begin{eqnarray*}
~_t p_{xy} &=& ~_t p_x times ~_t p_y = exp(-mu_1 t)exp(-mu_2 t) \
&=& exp(-(0.02+0.15)t) = exp(-0.035 t).
end{eqnarray*}
a) (~_t p_{xy})
begin{eqnarray*}
mu_{xy}(t) = mu_{x}(t)^m+mu_{y}(t)^f = mu_1 +mu_2 = 0.035.
end{eqnarray*}
b) ( mu_{xy}(t))
begin{eqnarray*}
~_t p_{overline{xy}} &=& ~_t p_x + ~_t p_y – ~_t p_x ~ ~_t p_y \
&=& exp(-mu_1 t) + exp(-mu_2 t) – exp(-(mu_1+mu_2) t) \
&=& exp(-0.02 t) + exp(-0.015 t) – exp(-0.035 t) .
end{eqnarray*}
c) (~_t p_{overline{xy}})
Exercise 3. Recall that we can express the Gompertz force of mortality as (mu_x = B c^x). In this case, we may express the survival function as ( S(x) = exp(-m(c^x-1))), where (m=B/ ln c) and the conditional survivor function for (T(x)) as
begin{eqnarray*}
~_t p_x = exp(-m(c^{x+t}-c^x))= exp(-m c^x(c^t-1)) .
end{eqnarray*} Assume that the Gompertz force governs mortality for males and females (with a common parameters (m) and (c)).
Determine the value of (w) so that we may write (~_t p_{xy} = ~_t p_w). That is, show that we can compute joint life probabilities using a single life table for the Gompertz case.
begin{eqnarray*}
~_t p_{xy} &=& ~_t p_x times ~_t p_y \
&=& exp(-m c^x(c^t-1)) exp(-m c^y(c^t-1))\
&=& exp(-( c^x + c^y) times m (c^t-1)) \
end{eqnarray*}
So, define (w) to be the solution of (c^w = c^x + c^y). With this, we may write
begin{eqnarray*}
~_t p_{xy} &=& exp(-( c^x + c^y) times m (c^t-1)) =exp(- m c^w (c^t-1))= ~_t p_w,
end{eqnarray*}
as required.