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3. Suppose you have reason to believe that you can predict the value of some random variable Y from the values of another random variable X (e.g. X is the number of classes a student has missed, and Y is their score on the final exam). In the simplest scenario, the predicted value of Y would be Ý = aX +b for some constants a and b (this would be called a linear predictor). The quality of this predictor can be measured using the mean square error E((Y – Ý )²). (a) Show that the mean square error is minimized if we pick constants b = p, a= µY – bµx , where p is the correlation coefficient between X and Y. (b) Calculate the mean square error for this best linear predictor. How does the value of p affect the error?

3. Suppose you have reason to believe that you can predict the value of some random variable Y from the values of another random variable X (e.g. X is the number of classes a student has missed, and Y is their score on the final exam). In the simplest scenario, the predicted value of Y would be Ý = aX +b for some constants a and b (this would be called a linear predictor). The quality of this predictor can be measured using the mean square error E((Y – Ý )²). (a) Show that the mean square error is minimized if we pick constants b = p, a= µY – bµx , where p is the correlation coefficient between X and Y. (b) Calculate the mean square error for this best linear predictor. How does the value of p affect the error?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 31EQ
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3. Suppose you have reason to believe that you can predict the value of some random variable Y from the values of another random variable X (e.g. X is the number of classes a student has missed, and Y is their score on the final exam). In the simplest scenario, the predicted value of Y would be Ý = aX +b for some constants a and b (this would be called a linear predictor). The quality of this predictor can be measured using the mean square error E((Y – Ý)²). (a) Show that the mean square error is minimized if we pick constants oy b = pY, a = HY – bµx , ox where p is the correlation coefficient between X and Y. (b) Calculate the mean square error for this best linear predictor. How does the value of p affect the error?
Transcribed Image Text:3. Suppose you have reason to believe that you can predict the value of some random variable Y from the values of another random variable X (e.g. X is the number of classes a student has missed, and Y is their score on the final exam). In the simplest scenario, the predicted value of Y would be Ý = aX +b for some constants a and b (this would be called a linear predictor). The quality of this predictor can be measured using the mean square error E((Y – Ý)²). (a) Show that the mean square error is minimized if we pick constants oy b = pY, a = HY – bµx , ox where p is the correlation coefficient between X and Y. (b) Calculate the mean square error for this best linear predictor. How does the value of p affect the error?
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