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A random variable X has unknown mean and variance o² = 25. To determine the mean u, one may approximate it by first generating n copies of independent random numbers X₁,..., Xn with the same distribution as X, then taking the statistical mean μl ≈ Xn = n²¹(X₁ ++ Xn). At least how large n need to be, in order to be 95% certain that the approximated mean X, is within an error of 0.01 from the actual value ? Find an estimation of n in two ways: (a) by Chebyshev's inequality; (b) by the central limit theorem. Compare the results. Particularly, which method give a better estimation?

A random variable X has unknown mean and variance o² = 25. To determine the mean u, one may approximate it by first generating n copies of independent random numbers X₁,..., Xn with the same distribution as X, then taking the statistical mean μl ≈ Xn = n²¹(X₁ ++ Xn). At least how large n need to be, in order to be 95% certain that the approximated mean X, is within an error of 0.01 from the actual value ? Find an estimation of n in two ways: (a) by Chebyshev's inequality; (b) by the central limit theorem. Compare the results. Particularly, which method give a better estimation?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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To A random variable X has unknown mean µ and variance ♂² = 25. determine the mean u, one may approximate it by first generating n copies of independent random numbers X₁,..., Xn with the same distribution as X, then taking the statistical mean µl ≈ Xn = n−¹(X₁ + ... + X₂). At least how large n need to be, in order to be 95% certain that the approximated mean X, is within an error of 0.01 from the actual value µ? Find an estimation of n in two ways: (a) by Chebyshev's inequality; (b) by the central limit theorem. Compare the results. Particularly, which method give a better estimation?
Transcribed Image Text:To A random variable X has unknown mean µ and variance ♂² = 25. determine the mean u, one may approximate it by first generating n copies of independent random numbers X₁,..., Xn with the same distribution as X, then taking the statistical mean µl ≈ Xn = n−¹(X₁ + ... + X₂). At least how large n need to be, in order to be 95% certain that the approximated mean X, is within an error of 0.01 from the actual value µ? Find an estimation of n in two ways: (a) by Chebyshev's inequality; (b) by the central limit theorem. Compare the results. Particularly, which method give a better estimation?
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