91Ó°ÊÓ

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In our scenario, where we're examining possible accident damages, each possible damage amount is an outcome, and each outcome has an associated probability.

For example, consider the damage values which are integrally tied to the characteristic variable, denoted as \(X\) in our case. The values of \(X\) are \(0, 1000, 5000,\) and \(10000\) dollars, representing the possible damages. The associated probabilities are:

• \(P(X=0) = 0.8\) implies an 80% chance of no damage occurring,
• \(P(X=1000) = 0.1\) signifies a 10% probability that the damage equals \(1000,
• \(P(X=5000) = 0.08\) indicates an 8% chance of a \)5000 damage event,
• \(P(X=10000) = 0.02\) represents a 2% likelihood of $10000 damages.

Understanding such formulations is crucial because they help assess risks and determine potential outcomes in uncertain environments, such as insurance calculations.

Insurance premium is the amount paid by the policyholder to maintain insurance coverage. Calculating the right premium amount is essential for insurance companies to ensure profitability while covering the risk. In the exercise, we work through the steps to find an appropriate premium amount to achieve a specific expected profit.

In general, the expected profit for the insurer is given by the equation: \[ \text{Premium} - \text{Expected Payment} = \text{Desired Profit} \] Here, the expected payment is the average financial outlay by the insurer, calculated using the expected value of potential payouts due to claims.
Given the deductible policy, certain damage values are absorbed by the policyholder. For damages \(X = 1000\), the policyholder covers the entire amount due to the deductible. Thus, the actual damages the insurer pays for \(X = 5000\) and \(X = 10000\) are \(4500\) and \(9500\) dollars respectively. With these conditions, the expected payout was calculated:

• 0 from \(X = 0\) and \(X = 1000\),
• \(360 from \(X = 5000\) (\(4500 \times 0.08\)),
• \)190 from \(X = 10000\) (\(9500 \times 0.02\)).

Thus, the expected payout totals to \(550. To achieve a desired profit of \)100, the premium should be $650, fulfilling the equation \( 650 - 550 = 100 \).

A deductible policy is a type of insurance policy where the policyholder is responsible for a specified amount of losses or damages before the insurer starts to pay. Deductibles are a common feature in many types of insurance and serve various purposes.

In this context, a \\(500 deductible means the policyholder covers the first \\)500 of any claim themselves. This impacts how insurance payouts are calculated:

• If the damages \(X = 1000\), the policyholder pays the full amount as it is below the deductible limit.
• For damages of \(5000\), the insurer pays \(4500\) (total damages minus the \\(500 deductible).
• Similarly, for damages of \(10000\), the payout by the insurer is \(9500\) after deducting \\)500.

The use of deductibles aligns interests between the insurer and policyholder by reducing the incentive for small claims and lowering the insurance cost, as evident in the calculated lower expected payout of \$550. This mechanism helps control costs for the insurer, translating to lower premiums for consumers.