Question

1. The daily rainfall in Dublin (measured in millimeters) is modelled using a gamma distribution with parameters α = 0.8 and β = 0.4.

(a) Write down the distribution (pdf) of the daily rainfall in Dublin.

(b) Use Markov’s inequality to upper bound the probability that the observed rainfall in a given day is larger than 3 mm, and compare the value to its true counterpart.

(c) Consider the overall rainfall in 365 days, and use mgfs and their properties to prove that this is Ga (292, 0.4).

(d) Use the central limit theorem to approximate the probability that the annual rainfall exceeds 800mm (write down the analytical formula and the code used to calculate the cdf value).

(e) Compare the approximate value obtained with the exact counterpart (include the R code used to calculate the cdf value).

(f) Use R to produce a graphical representation that compares the true and approximate densities (include both the code and plot).

Answer 1