Let ₁,...,, iid from N(u, o2) where o² is known. (a) Show that the Jeffreys prior for the normal likelihood is n(u) = C₁√n/0², μER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for u under the normal likelihood N(μ, o) 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 = 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 ₁,...,, iid from N(u, o2) where o² is known. (a) Show that the Jeffreys prior for the normal likelihood is n(u) = C₁√n/0², μER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for u under the normal likelihood N(μ, o) 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 = 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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