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:
Mathematics Assessmentfor, 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):
Mathematics Assessment
Mathematics Assessment Answer

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. This Mathematics Assignment has been solved by our Mathematics experts at TVAssignmentHelp. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing style. Be it a used or new solution, the quality of the work submitted by our assignment experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.

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