The cost of visits to physicians' offices follows an exponential distribution

Question

The cost of visits to physicians' offices follows an exponential distribution with an average of $250 per year per person. A health insurer sells a policy to 800 people with a $100 deductible. How much should the insurer expect to pay in benefits?

A $54,000

B $134,000

C $168,000

D $200,000

E $255,000

Answer 1

Let X be the total cost of visits to physicians' offices per year per person, and let Y be the amount paid by the health insurer per year per person.

Let's first compute the expected amount the health insurer will pay per year per person, E[Y].

The health insurer will not pay anything if the total cost of the visits for the entire year, X, is no greater than the deductible of $100. Otherwise, the health insurer pays the difference X-100. In other words, the random variable Y is defined by the following rule: it is equal to 0 if X ≤ 100, and equal to X-100 otherwise.

Furthermore, we can say that Y is a function of the random variable X, which we can write as Y = g(X), so the Law of the unconscious statistician applies and we obtain:

E[Y] = E[g(X)] = ∫_-∞^∞ g(x)f_X(x)dx = ∫_-∞^∞ (x-100)(1/250)e^-x/250dx ≈167.580, where f_X(x) = (1/250)e^-x/250 is the probability density function of X, which was assumed to be exponentially distributed with parameter λ = 1/250.

Thus, the expected amount in benefits to be paid to 800 people per year is ≈167.580*800 = 134,064.

So, answer choice B should be chosen.