Question 10 Suppose that we have data y = =(y,,yn). Each data point is assumed to be generated by a distribution with the following probability density function (pdf) py; n)=kny exp(-ny), y; ≥0,i=1,...,n. The unknown parameter is n, with k assumed to be known, and n > 0, k > 0. (a) Write the likelihood function for ŋ. Find an expression for the maximum likelihood estimate (MLE), n, for n. (b) A Gamma distribution, Gamma(a,b), is chosen as the prior for n with pdf p(n) bana-1 exp(-bn) F(a) η > 0, where a > 0 and b > 0 are known. The Gamma distribution for n is conjugate prior for the likelihood function in (a). (c) What does it mean to say that a Gamma(a, b) density for n is conjugate for the likelihood? (d) Derive the posterior distribution, p(ny), for n given the data y. (e) We would like to choose the Gamma prior distribution parameters such that the prior mean is, where B is the second-to-last digit of your ID number and the prior coefficient of variation (standard deviation divided by the mean) is 0.5. Find the prior Gamma distribution, Gamma(a, b), for n. (f) The data are y = (2,7,5,3,C+1), where C is the last digit of your ID number with n = 5. Set k = 2. (i) What is the MLE, n? (ii) Using the prior distribution from part (e), what is the the posterior distribution for n? Calculate the posterior mean.
Question 10 Suppose that we have data y = =(y,,yn). Each data point is assumed to be generated by a distribution with the following probability density function (pdf) py; n)=kny exp(-ny), y; ≥0,i=1,...,n. The unknown parameter is n, with k assumed to be known, and n > 0, k > 0. (a) Write the likelihood function for ŋ. Find an expression for the maximum likelihood estimate (MLE), n, for n. (b) A Gamma distribution, Gamma(a,b), is chosen as the prior for n with pdf p(n) bana-1 exp(-bn) F(a) η > 0, where a > 0 and b > 0 are known. The Gamma distribution for n is conjugate prior for the likelihood function in (a). (c) What does it mean to say that a Gamma(a, b) density for n is conjugate for the likelihood? (d) Derive the posterior distribution, p(ny), for n given the data y. (e) We would like to choose the Gamma prior distribution parameters such that the prior mean is, where B is the second-to-last digit of your ID number and the prior coefficient of variation (standard deviation divided by the mean) is 0.5. Find the prior Gamma distribution, Gamma(a, b), for n. (f) The data are y = (2,7,5,3,C+1), where C is the last digit of your ID number with n = 5. Set k = 2. (i) What is the MLE, n? (ii) Using the prior distribution from part (e), what is the the posterior distribution for n? Calculate the posterior mean.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 10
Suppose that we have data y = =(y,,yn). Each data point is
assumed to be generated by a distribution with the following probability density function (pdf)
py; n)=kny exp(-ny), y; ≥0,i=1,...,n.
The unknown parameter is n, with k assumed to be known, and n > 0, k > 0.
(a) Write the likelihood function for ŋ. Find an expression for the maximum likelihood
estimate (MLE), n, for n.
(b) A Gamma distribution, Gamma(a,b), is chosen as the prior for n with pdf
p(n)
bana-1 exp(-bn)
F(a)
η > 0,
where a > 0 and b > 0 are known. The Gamma distribution for n is conjugate prior for
the likelihood function in (a).
(c) What does it mean to say that a Gamma(a, b) density for n is conjugate for the
likelihood?
(d) Derive the posterior distribution, p(ny), for n given the data y.
(e) We would like to choose the Gamma prior distribution parameters such that the prior
mean is, where B is the second-to-last digit of your ID number and the prior
coefficient of variation (standard deviation divided by the mean) is 0.5. Find the prior
Gamma distribution, Gamma(a, b), for n.
(f) The data are y = (2,7,5,3,C+1), where C is the last digit of your ID number with
n = 5. Set k = 2.
(i) What is the MLE, n?
(ii) Using the prior distribution from part (e), what is the the posterior distribution for
n? Calculate the posterior mean.
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