Chapter 16, Problem 18E

(a)

To determine

To construct: a probability model for the gain on a policy.

Explanation of Solution

Given:

Price of the insurance policy = $100.

When a major injury happens, the amount paid by the insurance provider to the purchaser is $10,000. When a minor injury happens, the amount paid by the insurance provider to the purchaser is $3,000. The company calculates that every year 1 in every 2000 might have a major injury. The company calculates that every year 1 in every 500 might have a minor injury.

Calculation:

The probability a policyholder

For the major injury Is,

  $\begin{array}{c}P\left(\text{Major injury}\right)=\frac{1}{2000}\\ =0.0005\end{array}$

For minor injury is

  $\begin{array}{c}P\left(\text{Minor injury}\right)=\frac{1}{500}\\ =0.0020\end{array}$

The probability that company would have gain is,

  $\begin{array}{c}P\left(\text{Profit}\right)=1-P\left(\text{Major injury}\right)-P\left(\text{Minor injury}\right)\\ =1-0.0005-0.0020\\ =0.9975\end{array}$

Let X be the gain of the company.

Price of the Insurance policy is $100.

If the policyholder is major hurt, the amount the organization would recover from is,

  $\begin{array}{c}X=100-10000\\ =-9900\end{array}$

If the policyholder is minor hurt, the amount the organization would recover from is

  $\begin{array}{c}X=100-3000\\ =-2900\end{array}$

the probability distribution of gain on the policy is,

Type	X($)	P(X=x)
Major injury	-9900	0.0005
Minor injury	-2900	0.0020
Profit	100	0.9975
	Total	1

(b)

To determine

To find: the company’s predicted gain on this policy.

Answer to Problem 18E

$89

Explanation of Solution

Given:

From the part (a)

the probability distribution of gain on the policy is,

Type	X($)	P(X=x)
Major injury	-9900	0.0005
Minor injury	-2900	0.0020
Profit	100	0.9975
	Total	1

Formula used:

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

Calculation:

Expected gain is

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum xP\left(x\right)}\\ =\left(-9900\right)\left(0.0005\right)+\left(-2900\right)\left(0.0020\right)+100\left(0.9975\right)\\ =-4.95-5.80+99.75\\ =89\end{array}$

Therefore, the company’s predicted gain on the policy is $89.

(c)

To determine

To Calculate: the standard deviation.

Answer to Problem 18E

260.536

Explanation of Solution

Given:

From the part (a)

the probability distribution of gain on the policy is,

Type	X($)	P(X=x)
Major injury	-9900	0.0005
Minor injury	-2900	0.0020
Profit	100	0.9975
	Total	1

Formula used:

  $\begin{array}{c}E\left(X^2\right)={\displaystyle \sum x^{2}P\left(x\right)}\\ V\left(X\right)=E\left(X^2\right)-{\left[E\left(X\right)\right]}^2\\ \sigma =\sqrt{V\left(X\right)}\end{array}$

Calculation:

Estimating the standard deviation.

  $\begin{array}{c}E\left(X^2\right)={\displaystyle \sum x^{2}P\left(x\right)}\\ ={\left(-9900\right)}^2\left(0.0005\right)+{\left(-2900\right)}^2\left(0.0020\right)+{\left(100\right)}^2\left(0.9975\right)\\ =49005+16820+9975\\ =75800\end{array}$

The variance

  $\begin{array}{c}V\left(X\right)=E\left(X^2\right)-{\left[E\left(X\right)\right]}^2\\ =75800-{\left(89\right)}^2\\ =75800-7921\\ =67879\end{array}$

The standard deviation

  $\begin{array}{c}\sigma =\sqrt{V\left(X\right)}\\ =\sqrt{67879}\\ =260.536\end{array}$

Thus, the standard deviation of the gain on a policy is 260.536.