Suppose that random variable X 0, representing the annual rainfall (in metres) at your weather station, has a Gamma distribution, i.e., X Gamma(, ), where and are

positive parameters. Below, we re-parameterize by using the parameter = 1/.

(a) Use Python to describe your rainfall data series, by computing summary statistics, and plotting the kernel density function. Explain and discuss these results, including the nature and implications of the empirical distribution for rainfall at your weather station.

(b) Estimate the parameters of the Gamma distribution by the method of maximum likelihood, presenting your results (parameter estimates, standard errors, etc.) in tabular form.

(c) Plot the estimated (using the maximum likelihood estimates) Gamma density for

rainfall. Using the estimated Gamma density for rainfall, compute estimates of the probabilities of rainfall being (i) less than 70% of average annual rainfall (very dry), (ii) more than 130% of average annual rainfall (very wet) and (iii) either (i) or (ii). Discuss these results and their implications.

(d) Use the likelihood ratio test procedure to test the null hypothesis that = 1 (meaning

that the population mean and variance are equal) against the alternative hypothesis that = 1 using a 5% significance level. Your answer should provide full details of the logic used and your conclusion. [Hint: To estimate the model with = 1, replace '{theta}' by '1' in the estimation command.]

Include your Python program(s) at the end of your answers to this question.

The data:

Year	Rainfall16
1966	0.86043618
1967	0.78498955
1968	2.7866608
1969	2.7167775
1970	2.3972599
1971	0.38684201
1972	2.6825074
1973	5.0955186
1974	4.4039672
1975	2.1769829
1976	1.899169
1977	1.1354679
1978	1.54926
1979	1.2363492
1980	2.4929891
1981	6.2224455
1982	1.2718447
1983	3.7862099
1984	1.3253747
1985	0.93920953
1986	1.3986398
1987	1.1841027
1988	0.98393821
1989	1.8156087
1990	3.1067913
1991	1.0512532
1992	0.94449276
1993	0.4766173
1994	1.7962279
1995	1.0900812
1996	0.86254085
1997	3.4354329
1998	0.52220215
1999	3.4198457
2000	1.3095363
2001	1.7745219
2002	1.9668938
2003	5.6520617
2004	1.6694899
2005	3.1429521
2006	0.66600744
2007	1.0695483
2008	3.1826559
2009	0.52570474
2010	5.8281976
2011	0.83969946
2012	0.71746577
2013	0.91720641
2014	2.9881486
2015	3.5251173