Answer Happy

1. = The daily rainfall in Paris (measured in millimeters) is modelled using a gamma distribution with shape a = 1 and rate B = 0.5. It is assumed that the daily rainfall measurements are independent and identically distributed. (a) Find the numerical values of the mean, variance and standard deviation for the daily rainfall in Paris (deriving the formulas for these moments is not required) [5] (b) The amount of rainfall recorded for a given week in Paris was a {0.41, 0.23, 2.39, 2.02, 0.03, 1.28, 1.26, 1.06} Calculate the observed values of the sample mean and of the sample variance estimators. Do these underestimate or overestimate the mean and the variance of the population, respectively? [5] (c) Use Markov's inequality to attain an upper bound on the probability that the observed rainfall in a given day is larger than 5 mm. [6] (d) Prove that the annual rainfall over 365 days is a Gamma (365, 0.5). [7] (e) Use the central limit theorem to approximate the probability that the annual rainfall exceeds 750mm. [7]