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3. In class, we provided an example of deriving a two-sided confidence interval for 3 given a single observation Y~ Exp(3), with 1 fy(y): -y/B = y ≥ 0 andß> 0. Now suppose that we have n independent observations Y; ~ Exp(ß), and that we want to construct a one-sided confidence interval for B. In this exercise you may use the fact that Σ₁Y; ~ Gamma(n, B). i=1 (a) Show that Ỹ, the sample mean of the Yį, is a consistent estimator of ß. (b) Determine the sampling distribution of Y. As Y is a linear function of the n data Y₁,..., Yn, it is easiest to do this using the method of moment-generating functions. (c) Propose a pivot r.v. X that is a function of Y and ß (and possibly n) such that the pdf of X does not depend on 3. (d) Now set n = 3 and construct a one-sided confidence interval such that P(ß ≤ Âu) : = 0.95.

3. In class, we provided an example of deriving a two-sided confidence interval for 3 given a single observation Y~ Exp(3), with 1 fy(y): -y/B = y ≥ 0 andß> 0. Now suppose that we have n independent observations Y; ~ Exp(ß), and that we want to construct a one-sided confidence interval for B. In this exercise you may use the fact that Σ₁Y; ~ Gamma(n, B). i=1 (a) Show that Ỹ, the sample mean of the Yį, is a consistent estimator of ß. (b) Determine the sampling distribution of Y. As Y is a linear function of the n data Y₁,..., Yn, it is easiest to do this using the method of moment-generating functions. (c) Propose a pivot r.v. X that is a function of Y and ß (and possibly n) such that the pdf of X does not depend on 3. (d) Now set n = 3 and construct a one-sided confidence interval such that P(ß ≤ Âu) : = 0.95.

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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3. In class, we provided an example of deriving a two-sided confidence interval for 6 given a single observation Y ~ Exp(3), with 1 = fy(y) = -y/B B y ≥ 0 andß > 0. Now suppose that we have n independent observations Y;~ Exp(ß), and that we want to construct a one-sided confidence interval for B. In this exercise you may use the fact that Σ₁ Y; ~ Gamma(n, ß). =1 (a) Show that Y, the sample mean of the Y;, is a consistent estimator of 3. (b) Determine the sampling distribution of Y. As Y is a linear function of the n data Y₁,..., Yn, it is easiest to do this using the method of moment-generating functions. (c) Propose a pivot r.v. X that is a function of Y and ß (and possibly n) such that the pdf of X does not depend on 6. (d) Now set n = 3 and construct a one-sided confidence interval such that P(ß ≤ Ôu) 0.95. =
Transcribed Image Text:3. In class, we provided an example of deriving a two-sided confidence interval for 6 given a single observation Y ~ Exp(3), with 1 = fy(y) = -y/B B y ≥ 0 andß > 0. Now suppose that we have n independent observations Y;~ Exp(ß), and that we want to construct a one-sided confidence interval for B. In this exercise you may use the fact that Σ₁ Y; ~ Gamma(n, ß). =1 (a) Show that Y, the sample mean of the Y;, is a consistent estimator of 3. (b) Determine the sampling distribution of Y. As Y is a linear function of the n data Y₁,..., Yn, it is easiest to do this using the method of moment-generating functions. (c) Propose a pivot r.v. X that is a function of Y and ß (and possibly n) such that the pdf of X does not depend on 6. (d) Now set n = 3 and construct a one-sided confidence interval such that P(ß ≤ Ôu) 0.95. =
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