Chapter 5: Calculus and Probability

1. Explain a toy model for insurance that includes a deductible \(d\) and cap \(u\) in terms of how much the insurance company and customer each have to pay for a claim of size \(X\).
2. Define the total loss, payment, and uncovered loss in the context of a toy model for insurance.
3. Express the payment \(Y\) as a function of the total loss \(X\) when a plan:
   • has a deductible \(d\) and no cap
   • has a cap \(u\) and no deductible
   • has a deductible \(d\) and a cap \(u\)
4. Express the uncovered cost \(Z\) as a function of the total loss \(X\) when a plan:
   • has a deductible \(d\) and no cap
   • has a cap \(u\) and no deductible
   • has a deductible \(d\) and a cap \(u\)
5. Relate the total loss, payment, and uncovered loss algebraically.
6. Compute the expected value of the total loss, payment, and uncovered cost given a distribution for the total loss.
7. Explain why the payment and uncovered loss can be mixed-type random variables when the total loss is a continuous random variable.
8. Define the moment generating function \(M_{X}(t)\) for a random variable \(X\) as an expectation and in terms of its probability mass / density function.
9. Compute the \(k\)-th non-central moment \(E[X^{k}]\) of a random variable \(X\) using its moment generating function.
10. Given the moment generating function \(M_{X}(t)\) for a random variable \(X\), determine the moment generating function \(M_{Y}(t)\) for an affine transformation \(Y = a X + b\) of \(X\).
11. Given the moment generating functions \(M_{X}(t)\) and \(M_{Y}(t)\) for independent random variable \(X\) and \(Y\), determine the moment generating function \(M_{Z}(t)\) for their sum \(Z = X + Y\).
12. Use L’Hopital’s rule to evaluate a moment generating function or its derivatives at a removable discontinuity at \(t = 0\).