Question

Mr. Smith goes to his doctor and is told that he just tested positive for some fatal and rapidly progressing disease. Patients with this disease have a life expectancy exponentially distributed with mean 1 year. Patients of Mr. Smith's age without this disease have a life expectancy exponentially distributed with mean 20 years. 0.1% of the population (i.e., one person in a thousand) suffers from this disease. There is no known treatment that can alter its progression.

This test (like all medical tests) is not perfectly reliable: there is a 0.01 (i.e., 1%) probability that the test comes back positive for a healthy person. The probability that the test comes back negative for a person with this disease is also 0.01.

(a) What is the probability that Mr. Smith really does have this disease?
(b) Mr. Smith decides, at this stage, to buy life insurance. He wants a policy that will pay his family $1,000,000 if he dies within a year. If he is still alive after a year, his insurance policy expires and is worthless to him. The policy has a single, one-time, premium, of $100,000. Calculate the mean and variance of the profit (or loss) that the insurance company will make on this policy. Show all the steps in obtaining your result.
(c) Two years have passed since the initial test result and Mr. Smith is still alive! What is the probability
that he will be alive for at least another year (i.e., for at least 3 years following his initial diagnosis)? (You can assume that Mr. Smith catches no other disease during this two- year period.)

Answer 1

a)the probability that Mr Smith really does have this disease given that he tested positive =

b) life expectancy is exponentially distributed with the

probability that Smith will die within 1 year is

If Smith dies within 1 year, the insurance company will have a loss. Expected loss =

If Smith lives after 1 year, the insurance company will have a profit. Expected profit =

Hence on the total company will face an expected loss of 568890 -36790 = 532100

c) The probability that Smith will be alive at least 3 years following his initial diagnosis: