Posterior Probability and Prior Density - Equal-Tailed Bayesian Credible Interval - R-Code Assignment Help

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TASK: Question 1: The sequence (a n ) is defined by a 1 = 0 a 2 = 1 a n = a n?1 ? a n?2 + n for n ? 3 1. Write R codes to find a 100 . 2. The sequence (b t ) for t = 1, 2, .., 100 is given by: for, i = 1, 2, 3, ..., 100 Find mean, variance, standard deviation of b 1 , ..., b 100 . 3. Let Y ? P oisson(?). Suppose that we believe there are 100 possible values for ? = {b 1 , b 2 , ..., b 100 }. Assuming that your prior probability is equal across possible values of ?. Suppose Y= 1 was observed. 3.1 Calculate the posterior density. 3.2 What b i leads to highest posterior density? 4. Write the function f (x): Question 2: The number of buses that arrives for every 10 minutes follows a Poisson(?). Data shows that the number of buses for every 10 minutes over a period of 100 minutes are: {5,8,4,6,8,6,6,5,6,4}. 1. Your prior belief is uniform. 1.1 Calculate the posterior mean and variance. 1.2 Plot both posterior and prior density on the same graph. 2. Your prior belief about ? is that it has mean 6 and standard deviation 2. 2.1 Find a gamma(r, v) prior that matches your prior belief (Write a function to calculate r and v, given the mean and standard deviation) 2.2 What is the prior probability P (? > 7)? P (? < 4)? 2.3 Calculate the posterior mean and variance. 2.4 What is the posterior probability P (?|data > 7)? P (?|data < 4)? 2.5 Find the lower end of a 95% equal-tailed Bayesian credible interval for ? using both gamma distribution and normal approximation. 2.6 Plot both posterior and prior density on the same graph. 2.7 Prove that “Analyzing the observations sequentially one at a time” results in the same posterior distribution as “Analyzing the observations all together in a single step”. • We know that using the observations all together in a single step, the posterior P distribution is gamma(r + y i , v + n) (gamma(r, v) prior). • Write a function to obtain the posterior parameters with a single observation. Then write a loop to update each observation, take the previous posterior parameters to be the next prior parameters.

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