Consider a random sample X1, . . . , Xn of size n ≥ 2 from the normal distribution N (0, σ2 ) with parameter σ2 > 0 Suppose that the prior distribution for σ2 is inverse Gamma with density 11T(y; a, B) =r(a)Ba ya+1ey > 0with some positive constants a > 0 and 3 > 0 (this is indeed a probability density,i.e., you can use, without proof, that it integrates to 1 for any a, ß > 0).(1) Find the posterior distribution of o².(2) Find the Bayes estimator of o², that is, the mean of the posterior distribution.(3) Express the Bayes estimator from part (2) in terms of the MLE of o².

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Answered: 1 1 -南, r(@)3ª ya+1" т(у; а, B) y > 0…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Consider a random sample X1, . . . , Xn of size n ≥ 2 from the normal distribution N (0, σ2 ) with parameter σ2 > 0 Suppose that the prior distribution for σ2 is inverse Gamma with density

1
1
T(y; a, B) =
r(a)Ba ya+1e
y > 0
with some positive constants a > 0 and 3 > 0 (this is indeed a probability density,
i.e., you can use, without proof, that it integrates to 1 for any a, ß > 0).
(1) Find the posterior distribution of o².
(2) Find the Bayes estimator of o², that is, the mean of the posterior distribution.
(3) Express the Bayes estimator from part (2) in terms of the MLE of o².

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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