Tropical cyclones are classified based on maximum sustained wind speeds. In the Atlantic Ocean, there are three classes of tropical cyclones, as follows: - Tropical depressions have maximum sustained winds of less than 63 km/hr - Tropical storms have maximum sustained winds of 63-119 km/hr - Hurricanes have maximum sustained winds of 119 km/hr or more Let X be a random variable representing the maximum sustained winds of Atlantic tropical cyclones, that follows the Gamma distribution with the mean of 124 km/hr and the standard deviation of 53 km/hr. Assume that the maximum sustained wind speeds of tropical cyclones are independent of each other.

(a) Calculate the parameters of the distribution. (b) How high must the maximum sustained winds be for an Atlantic tropical cyclone to be among the highest 10%? (c) What proportion of Atlantic tropical cyclones are classified as hurricanes? (d) Suppose a particular tropical season has 16 tropical cyclones. Let H be the number of hurricanes in the tropical season. What is the distribution of H? Justify your answer. (e) Find the probability that at least half of the tropical cyclones are classified as hurricanes.

(f) Suppose a particular tropical season has 7 hurricanes. Let A be the number of tropical cyclones required in the tropical season to observe 7 hurricanes. What is the distribution of A? Justify your answer. (h) Find the probability that at least 17 tropical cyclones form for 7 hurricanes to be observed.