Consider the Bayesian estimation of the parameter 8 of a Poisson sample with a Gamma prior....

  

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Consider the Bayesian estimation of the parameter 8 of a Poisson sample with a Gamma prior. Assume that the true 0 = 2 and that the prior is r(3, 1). Write a Jupyter notebook with the following Monte Carlo simulation. Try different draws and check how your results change but you only have to report the results of one simulation (A) Plot the PDF of the prior (3, 1) clistribution. (b) Plot the PDF of Poisson distributions with 8 set to the prior mean, the prior mean plus 1 standard deviation of the prior, and the prior mean minus 1 standard deviation. (c) Generate a sample of the data of size 500, i.e. draw 500 data points from the Poisson distribution with 8 = 2. Assume that you observe the 500 observatious sequentially. For each i = 1,.., 500, compute the posterior distribution. Plot the parameters of the i-th posterior as well as the posterior mean and standard deviation. (d) Por each i = 1, ..., 500, compute and plot the mean and standard deviation of posterior distribution (e) Plot the posteriors for 1 = 5, 10, 20, 50,100,500. (f) For each i = 1...500, compute and plot the Bayes estimator for the quadratic loss function. Also plot the 90%, 95%, and 99% confidence intervals around the Bayes estimator (g) Derive the maximum like hood estimator, its standard error, and limiting distribution for the Poisson sample. (h) For each i = 1. ... 500, compute and plot the MLE, its standard deviation, and the 90%, 95%, and 99% confidence intervals. (i) Compare the Bayes and ML estimators and their distributions. Consider the effect of the sample size Consider the Bayesian estimation of the parameter 8 of a Poisson sample with a Gamma prior. Assume that the true 0 = 2 and that the prior is r(3, 1). Write a Jupyter notebook with the following Monte Carlo simulation. Try different draws and check how your results change but you only have to report the results of one simulation (A) Plot the PDF of the prior (3, 1) clistribution. (b) Plot the PDF of Poisson distributions with 8 set to the prior mean, the prior mean plus 1 standard deviation of the prior, and the prior mean minus 1 standard deviation. (c) Generate a sample of the data of size 500, i.e. draw 500 data points from the Poisson distribution with 8 = 2. Assume that you observe the 500 observatious sequentially. For each i = 1,.., 500, compute the posterior distribution. Plot the parameters of the i-th posterior as well as the posterior mean and standard deviation. (d) Por each i = 1, ..., 500, compute and plot the mean and standard deviation of posterior distribution (e) Plot the posteriors for 1 = 5, 10, 20, 50,100,500. (f) For each i = 1...500, compute and plot the Bayes estimator for the quadratic loss function. Also plot the 90%, 95%, and 99% confidence intervals around the Bayes estimator (g) Derive the maximum like hood estimator, its standard error, and limiting distribution for the Poisson sample. (h) For each i = 1. ... 500, compute and plot the MLE, its standard deviation, and the 90%, 95%, and 99% confidence intervals. (i) Compare the Bayes and ML estimators and their distributions. Consider the effect of the sample size