5. (S & W Chapter 2: 2.18) In any year, the weather can inflict storm damage to a home. From year to year, the damage is random. Let Y denote the dollar value of damage in any given year. Suppose that in 95% of the years Y=0 but in 5% of the years Y= $20,000. a) (5 points) What are the mean and standard deviation of the damage in any year? b) Consider an "insurance pool" of 100 people whose homes are sufficiently dispersed so that, in any year, the damage to different homes can be viewed as independently distributed random variables. Let I denote the average damage to these 100 homes in a year. (i) (5 points) What is the expected value of the average damage, P? (ii) (5 points) What is the probability that Y exceeds $2,000? 6. (S & W Chapter 3: 3.2) Let Y be a Bernoulli random variable with success probability Pr (Y=1)= p, and let Y1, Y2, ...., Y'n be i.i.d. draws from this distribution. Let p be the fraction of successes in this sample. a) (5 points) Show that p=Y. b) (5 points) Show that p is an unbiased estimator of p. c) (5 points) Show that var (p) = P(1-P)