Problem:
You have 100 business clients who own businesses that you insure against floods. The probability density function of the number of claims k per year for these 100 clients when rainfall is at or below average (which happens 50% of the time) is described by a binomial distribution in which floods are independent across the clients and the probability of a flood for each is 10% (p=.1) per year. Your insurance covers the first $100,000 in damage. For all your clients, if there is damage, it will very likely exceed $100,000. Thus, for simplicity assume that when there is a claim, you have to pay out $100,000 on each claim. You currently charge $20,000 per year for this coverage.
Required:
Question 1: When rainfall is at or below average, what is the expected number of claims?
Question 2: When rainfall is at or below average, what is the standard deviation of the number of claims?
Question 3: When rainfall is at or below average, what is the probability that you will have 5 or fewer claims? Show work.
Question 4: When rainfall is at or below average, what is the probability that you will lose money on your pool of policies? Show work.
Question 5: When rainfall is above average (which happens 50% of the time), the pdf for the number of claims is given by the binomial with p=.20. What is the probability that you will lose money if rainfall is above average?
Question 6: Given that rainfall is above average half the time and below average half the time, what is the unconditional probability that you will lose money if you charge $20,000 per $100,000 in coverage?
Please explain in detail and show all calculation and methods.