Question 4 (30 marks; 30 minutes) You met with four of your ECON3209 classmates to prepare...

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Question 4 (30 marks; 30 minutes) You met with four of your ECON3209 classmates to prepare for the Video assignment the day prior of the NSW state election. While discussing the assignment topic on frequentist and Bayesian inference, you decided to apply the two methods to predict the election's outcome for the Greens. You asked your peers to cast an anonymous vote of a 'Yes' for the Greens and a 'No' otherwise. You collected all the votes (including yours) and found two 'Yes' responses out of the five votes. (a) Frequentist inference: (i) Find maximum likelihood estimator of 0, the proportion of the Greens' voters in the NSW population of voters. (ii) Construct a 95% interval for 0. Note that here n is small, and thus you should use the Binomial distribution and not the CLT aproximation to construct the CI. (c) Assuming a flat uniform U[0, 1] prior for , (i) derive the posterior distribution of 0. Use Python (or any other software, including Excel) to plot the prior and posterior densities of given the number of 'Yes' voters in the sample. Comment on the result. (ii) compute the Bayes estimator of 0, defined as the expected value of the posterior density of 0. (ii) construct a 95% credible interval for 0. (c) Compare the two probabilistic intervals. Explain! Question 4 (30 marks; 30 minutes) You met with four of your ECON3209 classmates to prepare for the Video assignment the day prior of the NSW state election. While discussing the assignment topic on frequentist and Bayesian inference, you decided to apply the two methods to predict the election's outcome for the Greens. You asked your peers to cast an anonymous vote of a 'Yes' for the Greens and a 'No' otherwise. You collected all the votes (including yours) and found two 'Yes' responses out of the five votes. (a) Frequentist inference: (i) Find maximum likelihood estimator of 0, the proportion of the Greens' voters in the NSW population of voters. (ii) Construct a 95% interval for 0. Note that here n is small, and thus you should use the Binomial distribution and not the CLT aproximation to construct the CI. (c) Assuming a flat uniform U[0, 1] prior for , (i) derive the posterior distribution of 0. Use Python (or any other software, including Excel) to plot the prior and posterior densities of given the number of 'Yes' voters in the sample. Comment on the result. (ii) compute the Bayes estimator of 0, defined as the expected value of the posterior density of 0. (ii) construct a 95% credible interval for 0. (c) Compare the two probabilistic intervals. Explain!