Question

.Life insurance The risk of insuring one person’s life is reduced if we insure many people. Suppose that we insure two $21$-year-old males, and that their ages at death are independent. If $X_{1and}{X}_2$are the insurer’s income from the two insurance policies, the insurer’s average income W on the two policies is

$W=\frac{X_1+X_2}{2}=0.5X_1+0.5X_2$

Find the mean and standard deviation of $W$. (You see that the mean income is the same as for a single policy but the standard deviation is less.)

Short answer

Mean of W,${\mu }_W=303.35$,$\mu =303.35$

Standard deviation of W,${\sigma }_W=6864.29$

Step-by-step solution

Given Information

Given in the question that, the risk of insuring one person's life is reduced if we insure many people. Given that we insure two 21 year-old males and their ages at death are independent. Let $X_{1}and{X}_2$be the insurer's income from the two insurance policies. The insurer's average income W on the two policies is given by

$W=\frac{X_1+X_2}{2}=0.5X_1+0.5X_2$

We have to Find the mean and standard deviation of $W$.

Calculate the mean and standard deviation of W

Let's consider $X$as the random variable which shows the amount a life insurance company earns on a $5$year term life policy.

The mean and standard deviation of $X$are:

$$\begin{gathered} {\mu }_X=303.35 \\ {\sigma }_X=9707.57 \end{gathered}$$

Let's find the mean of $W$

localid="1649871962725" $$\begin{gathered} {\mu }_W=0.5{\mu }_{X_1}+0.5{\mu }_{X_2} \\ =0.5\times 303.35+0.5\times 303.35 \\ =303.35 \end{gathered}$$

Now, find the standard deviation of $W$

localid="1649872007295" $$\begin{gathered} {\sigma }_W=\sqrt{0.25\times 9707.{57}^2+0.25\times 9707.{57}^2} \\ =\sqrt{47118457.65} \\ =6864.29 \end{gathered}$$