Example: Life Expectancy

Suppose we believe the probability of dying before the age of

t0 is deﬁned by:

$\displaystyle P(0 \leq t < t_{0}) = \int_0^{t_{0}} \alpha(t)dt$

with,

$\displaystyle \alpha(t) = \left\{ \begin{array}{cc}

At^2 (100 - t)^2, &\mbox{if}\ 0 \leq t < 100\\

0 & \mbox{if}\ t \geq 100\end{array} \right.$

The condition,

$\displaystyle \int_0^\infty \alpha(t) dt = 1\ requires\ A = 3 * 10^{-9}$

Exercise: Verify this!

What is the probability of someone dying between 60 and 70

Answer

$\displaystyle P(60 \leq t < 70) = \int_{60}^{70} (3 * 10^{-9})t^2(100-t)^2 dt = 0.154$

Exercise: Verify this.

I cannot seem to get these answers correctly. Any help on how to do it?