Question

14. . Life insurance A life insurance company sells a term insurance policy to a $21$-year-old male that pays $\backslash (100,000$if the insured dies within the next $5$years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\backslash )250$each year a payment for the insurance. The amount $Y$that the company earns on this policy is $\backslash (250$per year, less the $\backslash )100,000$that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$profit earned by the company:

(a) Copy the table onto your paper. Fill in the missing values of $Y$.
(b) Find the missing probability. Show your work.
(c) Calculate the mean ${\mu }_Y$.Interpret this value in context

Short answer

(a) The missing value of $Y$table is:

Age of death	21	22	23	24	25	26 or more
Profit	-$99,750	-$99,500	-$99,250	-$99,000	-$98750	$1250
Probability	0.00183	0.00186	0.00189	0.00191	0.00193

(b) The missing probability is $0.99058$

(c) The mean is $303.3525$.

Step-by-step solution

Part (a) Step 1: Given Information

Given in the question that, a$21$-year-old male pays $\$100,000$If the insured dies within the next $5$years. And the company earns on this policy is $\$250$per year.

Part (a) Step 2: Explanation 

The client made $1$point of $\$250$for $21$years old

Therefore, the profit will be calculated as follow:

Profit = Revenue - Cost

$E\left(X\right)=\sum xp\left(x\right)$

So,

$$\begin{gathered} Profit=250-100000 \\ =-99750 \end{gathered}$$

The client made $2$payments of $\$250$for $22$years old:

Therefore, the payment will be:

$2\times 250=500$

Then, the profit will be calculated as follow:

$$\begin{gathered} Profit=Revenue-Cost \\ =500-100000 \\ =-99500 \end{gathered}$$

The client made $3$payments of $\$250$For $23$years old:

Therefore, the payment will be :

$3\times 250=750$

We can calculate the profit as follow

$$\begin{gathered} Profit=Revenue-Cost \\ =750-100000 \\ =-99250 \end{gathered}$$

The client made $4$payments of $\$250$for $24$years old:

Therefore, the payment will be:

$4\times 250=1000$

The profit will be calculated as follow

$$\begin{gathered} Profit=Revenue-Cost \\ =1000-100000 \\ =-99000 \end{gathered}$$

Part (a) Step 3: Create the table

The client made $5$payments of $\$250$for$25$ years old:
So, $5\times 250=1250$

The profit will be:

$$\begin{gathered} Profit=Revenue-Cost \\ =1250-100000 \\ =-98750 \end{gathered}$$

When the customer was $21$, he paid off his life insurance policy, and he had already made five payments of$\$250$ each (one payment per year).
So, the profit of the company is then the income of the $5$payments.

$$\begin{gathered} Profit=Revenue-Cost \\ =1250-0 \\ =1250 \end{gathered}$$

The complete table is:

Age of death	21	22	23	23	25	26 or more
Profit	-$99,750	-$99500	-$99250	-$99000	-$98750	$1250
Probability	0.00183	0.00186	0.00189	0.00191	0.00193

Part (b) Step 1: Given information 

We have to find the missing probability in the given table.

Part (b) Step 2: Explanation 

The property of probabilities is,

$\sum _{x}p\left(x\right)=1$
The probability values that are missing can be calculated as follows:

$$\begin{gathered} p(Y=-98750)=1-0.00183-0.00186-0.00189-0.00191-0.00193 \\ =0.99058 \end{gathered}$$

Part (c) Step 1: Given information 

The mean${\mu }_Y$to interpret the value in context.

Part (c) Step 2: Explanation 

The mean of $Y$is determined by:

$$\begin{gathered} E\left(Y\right)=\left[\begin{array}{c}(-99750)0.00183+(-99500)0.00186+(-99250)0.00189\\ +(-99000)0.00191+(-98750)0.00193+\left(1250\right)0.99058\end{array}\right] \\ =303.3525 \end{gathered}$$