Welcome to the tutorial on Exam C. Today we are going to go over Question #84, which is part
of the learning objective "Severity Models". Here is the view of the problem. Please choose one
of the answers and submit it. If incorrect hit the grey ‘x’ in the top right to try again. Here is the
solution to the question. Let X denote the number of hospital claims. According to the question,
X follows a Pareto distribution with parameters alpha=2 and theta=300. Let B denote the bonus
paid to the physicians. According to the question, B is equal to the amount by which the number
of claims is under 400 times c. With a formula, that is B=c(400-X). But apparently, this formula
only holds when the number of claims is lower than or equal to 400. When the number of claims
is above 400, there is no bonus, so B=0 in that case. What's also given is that the expected value
of the bonus is equal to 100. The question is asking for the value of c. The formula that we got
for the bonus has a piece-wise format. With a little algebra, we can re-organize the formulas and
get this. Please note that, only the second terms are different now. And we can actually write a
unique formula to denote B. That is, B=400c-c(X^400), with X^400 denoting the minimum
between X and 400. As given, the expected value of the bonus is equal to 100. We now have a
formula for the bonus B. So 100=E[400c-c(X^400)]. 400c is a constant number, so the expected
value of 400c is 400c itself. So we get the following equation. The question is asking for the
value of c. As long as we can calculate the limited expected value of (X^400), we'll be able to
solve for c. But how can we calculate this limited expected value? By definition, it can be
calculated with this integral, with f denoting the pdf of X. And this integral can be separated into
two parts, given by the following equation, with F denoting the cdf for the random variable X.
By using the density functions for Pareto distribution, we'll be able to find a formula for the
limited expected value. Alternatively, we can consult the Exam C Tables. It gives this formula
directly. If X follows a Pareto distribution with parameters alpha and theta, then the expected
value of the minimum of this random variable capital X and a constant number little x, can be
calculated by this formula, as long as alpha doesn't equal 1. For this question, alpha=2,
theta=300, and x=400. Plugin the numbers, we get the limited expected value is equal to 1200/7.
We plugin this number to the equation on the top, and we get the following equation.
That is, 100=400c-1200c/7. This is an equation about c, and c is the only unknown. So we can solve for
c from this equation. We get c=0.44, that's anwser A. That's the end of this tutorial. Thanks for watching!