I. Calculations for a Discrete Distribution - Expected Value and Standard Deviation Consider the five level system for rating the intensity of hurricanes and tornados. We want to assess and predict what the Florida Gulf Coast and the High Plains Midwest will experience in the coming year. Define the random variable: X = intensity of a hurricane or tornado - F-Level or Category The probability distribution for this random variable is given as follows: Xi P(Xi) 1 25% = .25 2 30% = .30 3 20% = .20 4 15% = .15 5 10% = .10 [Q#1] What is the Expected Value for X - the mean intensity level for these natural disasters? (a) 1.25 (b) 1.85 (c) 2.50 (d) 2.55 (e) 2.95 [Q#2] What is the Standard Deviation value for this distribution? (a) 1.284 (b) 1.444 (c) 1.6475 (d) 1.777 (e) 2.00 II. Expected Monetary Value and Discrete Random Variables Life Insurance Consider the situation for a life insurance company which has experienced a major shift in its business model due to the Coronavirus pandemic. The mortality rate for its insured clients has soared to: P(Death) = 0.005. At the same time, clients now want twice as much insurance with the average term life insurance policy at: $300,000. This is the payout if death occurs. The company has been able to raise the Fee it charges substantially to: F = $1,000 . Complete the table below that incorporates these values for event outcomes and their probability values. Complete the table and answer the questions. Event: Payout(Xi) P( Xi ) _________ X = Live $ X = Die $ [Q#1] What is the expected payout per insured customer? (a) $15 (b) $150 (c) $500 (d) $1,000 (e) $1,500 [Q#2] What is the expected profit per insured customer? (a) -$500 (b) $0 (c) $500 (d) $1,000 (e) $1,500 Homeowners Insurance A major insurance company - State Farm - provides homeowners insurance to n = 1,000,000 customers. Payouts based upon damage caused by fires, wind and flooding can be grouped into four categories based on damage intensity. The following table provides the outcome and probability information that guides the price/fee setting policy for State Farm Insurance. Damage Payout Probability Intensity Xi P(Xi) __________ None $0 93.9% = .939 Minor $3,000 5.0% = .05 Major $30,000 1.0% = .01 Catastrophic $300,000 .1% = .001 [Q#1] What is the expected payout per customer? (a) $150 (b) $300 (c) $500 (d) $750 (e) $900 [Q#2] If the insurance company receives payments from premiums that average F = $900 per customer then what is its operating profit margin for this set of n = 1,000,000 customers? (a) -$50 million (c) +$150 million (e) $900 million (b) $0 (d) +$300 million [Q#3] How many insured customers - from the n = 1,000,000 total - are expected to suffer damage at a level that is Major or Catastrophic? (Major or greater) (a) 11,000 (b) 10,000 (c) 1,000 (d) 0