Insurance Reimbursement An insurance policy reimburses a...

Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y,$ follows a distribution with density function: $$f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right.$$ What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries. $\begin{array}{lllllll}{\text { (a) } 1.0} & {\text { (b) } 1.3} & {\text { (c) } 1.8} & {\text { (d) } 1.9} & {\text { (e) } 2.0}\end{array}$

Insurance Reimbursement An insurance policy reimburses a loss up to a benefit limit of $10 .$ The policyholder's loss, $Y,$ follows a distribution with density function:
$$f(y)=\left\{\begin{array}{ll}{\frac{2}{y^{3}}} & {\text { for } y>1} \\ {0} & {\text { otherwise }}\end{array}\right.$$
What is the expected value of the benefit paid under the insurance policy? Choose one of the following. (Hint: The benefit paid will be equal to the actual loss if the actual loss is less than the limit. Otherwise it will equal the limit.) Source: Society of Actuaries.
$\begin{array}{lllllll}{\text { (a) } 1.0} & {\text { (b) } 1.3} & {\text { (c) } 1.8} & {\text { (d) } 1.9} & {\text { (e) } 2.0}\end{array}$

Key Concepts

Loss Distribution

A loss distribution describes the probability model of losses encountered by an insurer. It specifies the likelihood of different loss amounts using probability density or mass functions, which can be used to calculate metrics like the probability of a loss being above or below a certain threshold.

Expected Value

The expected value is the weighted average of all possible outcomes of a random variable, where the weights are the probabilities of those outcomes. In the context of continuous distributions, it is computed as an integral of the product of the loss value and its density function over the range of possible losses.

Policy Limit (Capped Benefit)

In insurance, a policy limit is the maximum amount that an insurer will pay on a covered loss. When computing the expected payment under such a policy, the actual loss is capped by the policy limit. This truncation means that for losses exceeding the limit, the insurer’s payment is fixed at the maximum benefit, which requires the expected value to be computed piecewise to account for losses both below and above the limit.

Integration in Continuous Distributions

Integration is a fundamental tool in probability for calculating the expected value and other properties of continuous random variables. It involves summing over an uncountably infinite set of outcomes by integrating the product of the value of interest (such as loss amount) and its probability density across the relevant range.

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