This problem explores the question of estimating (> 0) using a confidence interval based on a random sample X₁, X2, ..., Xn from the distribution with pdf fo(x) = { 5 exp(-) if=> 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating & using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a) % confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and x₁ = 1. : 1 the (e) Is the 95% confidence interval for o you obtained in (d) when n = = 1 and 21 shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.
This problem explores the question of estimating (> 0) using a confidence interval based on a random sample X₁, X2, ..., Xn from the distribution with pdf fo(x) = { 5 exp(-) if=> 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating & using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a) % confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and x₁ = 1. : 1 the (e) Is the 95% confidence interval for o you obtained in (d) when n = = 1 and 21 shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:This problem explores the question of estimating o(> 0) using a confidence interval
based on a random sample X1, X2, ..., Xn from the distribution with pdf
fo(x) =
{& exp(-
20
if x > 0
otherwise.
(a) Find the pdf of X2 by finding the cdf and differentiating it.
(b) Using the result of (a) and techniques we developed in class, propose a pivot for
estimating using a confidence interval and find the distribution of your pivot. Review
Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used
to construct a confidence interval.
(c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for
finding a short 100(1-a)% confidence interval for o.
(d) Using the formula obtained in (c), compute the width of the resulting 95% confidence
interval when n = 1 and ₁ = 1.
(e) Is the 95% confidence interval for a you obtained in (d) when n = 1 and ₁ = 1 the
shortest possible? If your answer is yes, explain why. If the answer is no, give a
counterexample.
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VIEWStep 2: Find the pdf of X1^2 by finding the cdf and differentiating it
VIEWStep 3: Propose a pivot for estimating σ and find the distribution of your pivot
VIEWStep 4: Write down the formula for finding a short 100(1-α)% confidence interval for σ.
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