.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.)

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 $W$.

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