The number of flooding events hitting a community next year can be 0, 1, 2 or 3, with probability 40%, 30%, 20% and 10%, respectively. Because of the flooding, the infrastructure serving the community can result in one of three states: "functioning", "partially functioning", or "malfunctioning". The conditional probability of these three states is 80%, 20% and 0 if no flooding occurs; 50%, 30% and 20% if one flooding event occurs; 30%, 40% and 30% if two events occur; 10%, 60% and 30% if three events occur, respectively. The overall annual maintenance cost is $5M, $15M and $50M, depending on the infrastructure being functioning, partially functioning, or malfunctioning, respectively. a) Find the marginal probability of the annual maintenance cost. b) What is the expected annual cost? What is the standard deviation of the annual cost? c) What is the expected annual cost conditional to the number of flooding events being 0, 1, 2 or 3, respectively? What is the corresponding standard deviation in these four cases? d) Verify the formula for total expectation and total variance, by comparing the results on question (a) and those of question (c): how can you derive the expectation and variance of the cost as a function of the conditional expectation and variance given the number of events? e) If the cost is $50M, what is the probability that 0, 1, 2 or 3 events have hit the community?

Answer should be typed format and don't insert any table and image.

Copied content strictly prohibited.