Question

Let X_1,...,X_n|theta ~(iid) N(theta,sigma^2) where sigma^2 is known. Let the prior distribution be Uniform, i.e., Theta~Unif(-inf,inf). a) Find the posterior distribution of theta given the data. b) Find the Bayes estimate assuming squared-error loss. c) Give the equal-tailed 95% credible interval. d) Given that sigma^2=4, n=10, and (sum from i=1 to 10 of x_i) = 80, calculate the length of the interval you found in the last part.

Name: let x1xntheta iid nthetasigma2 where sigma2 is known let the prior distribution be uniform ie thetaunif infinf a find the posterior distribution of theta given the data b find the bayes es 83204
Uploaded: 2024-01-11T01:10:41-08:00
Duration: 2 min
Channel: Nick Johnson
Description: let x1xntheta iid nthetasigma2 where sigma2 is known let the prior distribution be uniform ie thetaunif infinf a find the posterior distribution of theta given the data b find the bayes es 83204

          Let X_1,...,X_n|theta ~(iid) N(theta,sigma^2) where sigma^2 is known. Let the prior distribution be Uniform, i.e., Theta~Unif(-inf,inf).
a) Find the posterior distribution of theta given the data.
b) Find the Bayes estimate assuming squared-error loss.
c) Give the equal-tailed 95% credible interval.
d) Given that sigma^2=4, n=10, and (sum from i=1 to 10 of x_i) = 80, calculate the length of the interval you found in the last part.

Added by Jason H.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Nick Johnson

01/11/2024

Step 1

Given that X_1,...,X_n|theta ~(iid) N(theta,sigma^2) and the prior distribution of Theta is Uniform(-inf,inf), we need to find the posterior distribution of Theta given the data. The likelihood function for the data is given by: L(theta) = Show more…

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Let X_1,...,X_n|theta ~(iid) N(theta,sigma^2) where sigma^2 is known. Let the prior distribution be Uniform, i.e., Theta~Unif(-inf,inf). a) Find the posterior distribution of theta given the data. b) Find the Bayes estimate assuming squared-error loss. c) Give the equal-tailed 95% credible interval. d) Given that sigma^2=4, n=10, and (sum from i=1 to 10 of x_i) = 80, calculate the length of the interval you found in the last part.