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2. (a) Consider random variable X1, X2,..., X,. from a Poisson distribution with 2 = 2. State the probability density function (pdf) and its parameter(s). Compute the expected value and the variance of the random variable. (4 marks) b) Generate random sample of size n = 30 from the Poisson distribution with 2 = 2. Calculate the sample mean and sample standard deviation of the generated sample, give your answer in 3 decimal places. (5 marks) c) Calculate a 95% confidence interval based on sample generated in part b). Assuming it is normally distributed. (4 marks) d) Perform 1,000 repetitions from part (b). Plot the histograms and compute the E[ X ] and Var( X) . Comment on your findings in the context of the central limit theorem. (15 marks) e) Calculate a 95% confidence interval based on the 1,000 samples in part d). (4 marks) f) Compare the answer of confidence intervals in part c) and e). Which confidence interval do you prefer more? Justify your