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Let independent positive random variables X1, X2,..., X, be modelled by a gamma distribution with known parameter a = 2 and unknown parameter 0 > 0, so that for i = 1,2,...,n, f(ri|0) = 0²r, exp(-Or.) on ri > 0. %3D (a) Show that L(0), the likelihood for 6 based on observed data x = (r1 r2 In)", can be written as ... L(0) x 02" exp(-OnT). Explicitly mention any terms that have been removed from the core of the likelihood, and give the reason for removal. (b) Suppose that a Gamma(a, b) prior is specified for 0. Show that the core of the posterior f(0|æ) can be written as S(0\æ) x 0ª+2n-1exp{-(b+n7)0}. (c) Identify the posterior distribution for 6, and give its parameter value(s).

Let independent positive random variables X1, X2,..., X, be modelled by a gamma distribution with known parameter a = 2 and unknown parameter 0 > 0, so that for i = 1,2,...,n, f(ri|0) = 0²r, exp(-Or.) on ri > 0. %3D (a) Show that L(0), the likelihood for 6 based on observed data x = (r1 r2 In)", can be written as ... L(0) x 02" exp(-OnT). Explicitly mention any terms that have been removed from the core of the likelihood, and give the reason for removal. (b) Suppose that a Gamma(a, b) prior is specified for 0. Show that the core of the posterior f(0|æ) can be written as S(0\æ) x 0ª+2n-1exp{-(b+n7)0}. (c) Identify the posterior distribution for 6, and give its parameter value(s).

MATLAB: An Introduction with Applications
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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Let independent positive random variables X1, X2,..., Xn be modelled by a gamma distribution with known parameter a = 2 and unknown parameter 0>0, so that for i = 1, 2, ..., n, %3D f(r;|0) = 0²x¡ exp(-Or;) on r; > 0. %3D (a) Show that L(0), the likelihood for 0 based on observed data x = (r1 x2 In)", can be written as ... L(0) x 0²" exp(-On7). Explicitly mention any terms that have been removed from the core of the likelihood, and give the reason for removal. (b) Suppose that a Gamma(a, b) prior is specified for 6. Show that the core of the posterior f(0|x) can be written as f(0|x) x 0a+2n-'exp{-(b+nF)0}. (c) Identify the posterior distribution for 6, and give its parameter value(s).
Transcribed Image Text:Let independent positive random variables X1, X2,..., Xn be modelled by a gamma distribution with known parameter a = 2 and unknown parameter 0>0, so that for i = 1, 2, ..., n, %3D f(r;|0) = 0²x¡ exp(-Or;) on r; > 0. %3D (a) Show that L(0), the likelihood for 0 based on observed data x = (r1 x2 In)", can be written as ... L(0) x 0²" exp(-On7). Explicitly mention any terms that have been removed from the core of the likelihood, and give the reason for removal. (b) Suppose that a Gamma(a, b) prior is specified for 6. Show that the core of the posterior f(0|x) can be written as f(0|x) x 0a+2n-'exp{-(b+nF)0}. (c) Identify the posterior distribution for 6, and give its parameter value(s).
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