91Ó°ÊÓ

When discussing probability models, one of the key concepts is the expected value. This concept tells us what to anticipate on average over numerous trials of an event. For the insurance policy problem here, the expected value represents the average profit or loss per policy. We achieve this by multiplying the profit of each possible outcome by its respective probability, then summing them up.

To calculate it:

• For a major injury, the profit is -\(9,900, and it happens with a probability of \( \frac{1}{2000} \).
• For a minor injury, the profit is -\)2,900, and its probability is \( \frac{1}{500} \).
• If no injury occurs, the profit is $100 with a probability of \( \frac{219}{2200} \).

The expected value calculation is then:\[ E(X) = (-9900) \times \frac{1}{2000} + (-2900) \times \frac{1}{500} + (100) \times \frac{219}{2200} \]This result will give the insurance company the average profit it can expect from issuing this policy.

Standard deviation is a metric used to measure the amount of variation or dispersion in a set of values. In simpler terms, it tells you how much the profits in this insurance policy example will deviate from the expected value on average.

The calculation process involves first finding the variance, which is the average of the squared differences from the expected value:

• Calculate the squared profit for each outcome, multiplied by its probability: \((-9900)^2 \times \frac{1}{2000}\), \((-2900)^2 \times \frac{1}{500}\), and \((100)^2 \times \frac{219}{2200}\).
• Sum these values to get \( E(X^2) \).
• Subtract \((E(X))^2\) from \(E(X^2)\) to find the variance.

Finally, take the square root of the variance to get the standard deviation: \( \sqrt{\text{Var}(X)} \). This value helps the company understand the range of the policy's profit variability and assess the risk involved.

An insurance policy acts as a contract between the insurance company and the policyholder, entailing the payment of coverage in exchange for premiums. In this scenario, the insurance company offers a policy for $100. In the event of an injury, it compensates differently based on the severity.

This insurance setup serves several purposes:

• For policyholders, it provides financial protection against severe and minor injuries.
• For the insurance company, the policy intends to be profitable while managing the risk of multiple claims simultaneously.

The insurance company must estimate risks accurately, using mathematical models to predict profitability and establish how much to charge for premiums. This is where concepts like expected value and standard deviation play a critical role. They help assess the likelihood of potential payouts versus the collected premiums, ensuring the policy remains financially feasible.

Variance is another statistical measure that provides insight into the spread of a set of data points. In the context of our insurance policy problem, variance helps to understand how much individual outcomes differ from the expected profit on average.

To calculate variance, follow these steps:

• Find \(E(X^2)\), which is the expected value of profits squared, using each outcome's square of profit times its probability.
• Subtract the square of the expected profit \((E(X))^2\) from \(E(X^2)\).

Mathematically, variance is expressed as \( \text{Var}(X) = E(X^2) - (E(X))^2 \). It tells the company about the variability in profits from each policy, guiding decision-making by highlighting potential financial risks. A higher variance indicates more uncertainty and risk, an essential consideration for insurance firms when setting premium rates and designing policies.