Chapter 6.1, Problem 14E

a.

To determine

To complete: The provided table.

Explanation of Solution

Given:

A 21 years old male that pays $100000, if the insured dies within the next 5 year. And the amount that company earn is 250 per years.

Formula used:

The formula to compute the profit is:

  $\text{Profit}=\text{Revenue}-\text{Cost}$

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

Calculation:

Consider that n is defined as the number of years.

  $\text{Earn}=n\left(\text{\$250}\right)$

Each year company will earn,

For 1 year,

  $\begin{array}{c}\text{Profit}=250-100000\\ =-99750\end{array}$

For 2 year,

  $\begin{array}{c}\text{Profit}=500-100000\\ =-99500\end{array}$

For 3 year,

  $\begin{array}{c}\text{Profit}=750-100000\\ =-99250\end{array}$

For 4 year,

  $\begin{array}{c}\text{Profit}=1000-100000\\ =-99000\end{array}$

For 5 year,

  $\begin{array}{c}\text{Profit}=1250-100000\\ =-98750\end{array}$

For more than 5 year,

  $\begin{array}{c}\text{Profit}=1250-0\\ =1250\end{array}$

Thus, the complete table is:

Age of death	21	22	23	24	25	26 or more
Profit	-$99,750	-$99,500	-$99,250	-$99,000	-$98,750	$1250

b.

To determine

To compute: The missing probabilities.

Explanation of Solution

Calculation:

By using property of probabilities,

  ${\displaystyle {\sum }_{x}p\left(x\right)=1}$

The missing probabilities values, can be calculated as;

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

Thus, the required probability is 0.99058.

c.

To determine

To compute: The mean of Y .

Answer to Problem 14E

The mean of Y is 303.3525.

Explanation of Solution

Calculation:

The mean of Y can be computed as:

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

Thus, the required mean is 303.3525.