Let x₁,..., n iid from N(μ, o2) where o2 is known. (a) Show that the Jeffreys prior for the normal likelihood is p(μ) =₁₁√√√n/o², µER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for µ under the normal likelihood N(µ, o²) and Jeffreys prior for u. Plot the density. (d) Simulate 1,000 draws from the posterior derived in (c) and plot a histogram of the simulated values. (e) Let 0= exp(µ). Find the posterior density of analytically and plot the density. (f) Estimate by Monte Carlo integration. (g) Compute a 95% equal tail interval for analytically and by simulation.
Let x₁,..., n iid from N(μ, o2) where o2 is known. (a) Show that the Jeffreys prior for the normal likelihood is p(μ) =₁₁√√√n/o², µER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for µ under the normal likelihood N(µ, o²) and Jeffreys prior for u. Plot the density. (d) Simulate 1,000 draws from the posterior derived in (c) and plot a histogram of the simulated values. (e) Let 0= exp(µ). Find the posterior density of analytically and plot the density. (f) Estimate by Monte Carlo integration. (g) Compute a 95% equal tail interval for analytically and by simulation.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 7CR
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please answer part d,e,f,g only using r code showing the code and method
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VIEWStep 2: Simulate 1,000 draws from the posterior derived in (c) and plot a histogram of the simulate
VIEWStep 3: Find the posterior density of θ analytically and plot the density given θ = exp(μ)
VIEWStep 4: Estimate by Monte Carlo integration
VIEWStep 5: Compute a 95% equal tail interval for θ analytically and by simulation
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