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Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam does not know the value of , but has access to a random sample Y₁, Yio- Liam has decided to predict Y₁₁ using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁₁ is not part of the estimation sample Y₁Y₁o. It can be a useful exercise to figure out where in the calculation it makes a difference. (a) Compute the bias of the prediction. (b) Compute the mean of the prediction error. (e) Compute the variance of the prediction error. (d) Compute the MSPE of the prediction. (e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (f) Suppose μ = 10 (instead of μ=2). Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (g) In a realistic setting, the value of μ is unknown. What advice would you give someone who is deciding between using Y and Y/2?

Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam does not know the value of , but has access to a random sample Y₁, Yio- Liam has decided to predict Y₁₁ using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁₁ is not part of the estimation sample Y₁Y₁o. It can be a useful exercise to figure out where in the calculation it makes a difference. (a) Compute the bias of the prediction. (b) Compute the mean of the prediction error. (e) Compute the variance of the prediction error. (d) Compute the MSPE of the prediction. (e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (f) Suppose μ = 10 (instead of μ=2). Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (g) In a realistic setting, the value of μ is unknown. What advice would you give someone who is deciding between using Y and Y/2?

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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Parts, E, F, and G.

## Problem 1 Liam does not know the value of \(\mu\), but has access to a random sample \(Y_1, \ldots, Y_{10}\). Liam has decided to predict \(Y_{11}\) using \(Y/2\) instead of \(\bar{Y}\). **Tasks:** - **(a)** Compute the bias of the prediction. - **(b)** Compute the mean of the prediction error. - **(c)** Compute the variance of the prediction error. - **(d)** Compute the MSPE of the prediction. - **(e)** Does Liam’s prediction \(Y/2\) produce a prediction with a lower MSPE than Olivia’s \(\bar{Y}\) prediction? - **(f)** Suppose \(\mu = 10\) (instead of \(\mu = 2\)). Does Liam's prediction \(Y/2\) produce a prediction with a lower MSPE than Olivia's \(\bar{Y}\) prediction? - **(g)** In a realistic setting, the value of \(\mu\) is unknown. What advice would you give someone who is deciding between using \(Y\) and \(Y/2\)? ## For Reference: Exercise 14.5 \(Y\) is a random variable with mean \(\mu = 2\) and variance \(\sigma^2 = 25\). ### Tasks: - **(a)** Suppose Emma knows the value of \(\mu\). - **(i)** What is the best (lowest MSPE) prediction of \(Y\) that Emma can make? That is, what is the oracle prediction of \(Y\)? - **(ii)** What is the MSPE of Emma’s prediction? - **(b)** Suppose Olivia does not know the value of \(\mu\) but has access to a random sample of size \(n = 10\) from the same population (represented by variables \(Y_1, \ldots, Y_{10}\)). Let \(\bar{Y}\) denote the sample mean from this random sample. Olivia wants to predict \(Y_{11}\), which is not part of her sample \(Y_1, \ldots, Y_{10}\). Let us denote \(Y\) as \(Y_{11}\) to emphasize this. Olivia has decided to predict the value of \(
Transcribed Image Text:## Problem 1 Liam does not know the value of \(\mu\), but has access to a random sample \(Y_1, \ldots, Y_{10}\). Liam has decided to predict \(Y_{11}\) using \(Y/2\) instead of \(\bar{Y}\). **Tasks:** - **(a)** Compute the bias of the prediction. - **(b)** Compute the mean of the prediction error. - **(c)** Compute the variance of the prediction error. - **(d)** Compute the MSPE of the prediction. - **(e)** Does Liam’s prediction \(Y/2\) produce a prediction with a lower MSPE than Olivia’s \(\bar{Y}\) prediction? - **(f)** Suppose \(\mu = 10\) (instead of \(\mu = 2\)). Does Liam's prediction \(Y/2\) produce a prediction with a lower MSPE than Olivia's \(\bar{Y}\) prediction? - **(g)** In a realistic setting, the value of \(\mu\) is unknown. What advice would you give someone who is deciding between using \(Y\) and \(Y/2\)? ## For Reference: Exercise 14.5 \(Y\) is a random variable with mean \(\mu = 2\) and variance \(\sigma^2 = 25\). ### Tasks: - **(a)** Suppose Emma knows the value of \(\mu\). - **(i)** What is the best (lowest MSPE) prediction of \(Y\) that Emma can make? That is, what is the oracle prediction of \(Y\)? - **(ii)** What is the MSPE of Emma’s prediction? - **(b)** Suppose Olivia does not know the value of \(\mu\) but has access to a random sample of size \(n = 10\) from the same population (represented by variables \(Y_1, \ldots, Y_{10}\)). Let \(\bar{Y}\) denote the sample mean from this random sample. Olivia wants to predict \(Y_{11}\), which is not part of her sample \(Y_1, \ldots, Y_{10}\). Let us denote \(Y\) as \(Y_{11}\) to emphasize this. Olivia has decided to predict the value of \(
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