A husband and wife purchase life insurance policies.Over the next 40 years, one policy pays out when the husband dies, and the other pays out when both husband and wife die.Their life expectancy is 20 years, and the probability that both die before year t is given by the function $f_{T}(t)=\frac{1}{1600} t^{2}$
.How fast is the probability that both are dead increasing in 25 years?

Question

A husband and wife purchase life insurance policies.Over the next 40 years, one policy pays out when the husband dies, and the other pays out when both husband and wife die.Their life expectancy is 20 years, and the probability that both die before year t is given by the function $f_{T}(t)=\frac{1}{1600} t^{2}$
 .How fast is the probability that both are dead increasing in 25 years?

Quizplus · Accepted Answer

The rate of change of the probability that both are dead is given by the derivative of the function $f_{T}(t)$. Differentiating $f_{T}(t)=\frac{1}{1600} t^{2}$ with respect to $t$, we get $f'_{T}(t)=\frac{1}{800} t$. Substituting $t=25$ gives $f'_{T}(25)=\frac{1}{800} \times 25 = 0.03125$, which rounds to 0.0313.

A Husband and Wife Purchase Life Insurance Policies fT(t)=11600t2f_{T}(t)=\frac{1}{1600} t^{2}fT​(t)=16001​t2 How Fast Is the Probability That Both Are Dead Increasing

A Husband and Wife Purchase Life Insurance Policies $f_{T}(t)=\frac{1}{1600} t^{2}$ How Fast Is the Probability That Both Are Dead Increasing