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1. Theorem: Let X = (X₁, X2, ..., X₁)' be a r.s. from N(u), where μER and >0 and n ≥ 2. Then, a. X and S² are independent;

1. Theorem: Let X = (X₁, X2, ..., X₁)' be a r.s. from N(u), where μER and >0 and n ≥ 2. Then, a. X and S² are independent;

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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1. Theorem: Let X = (X₁, X2, ..., X₁)' be a r.s. from N(ud), where μER and >0 and n ≥2. Then, a. X and S² are independent; 2. Defn: A continuous r.v. X is said to have an F-distribution with m (numerator) and n (denominator) degrees of freedom (d.f.) if, and only if, the PDF of X is given by Notation: Mean # Variance: T() rr) (1+x)(x), m.nez £x(x) = (m) ² X-F) E(X) = -²²2: "> n>2 Var(X) = DNE 2n² (m+n-2) m(n-2)(n-4) n>4 Theorem: Let U-X and V-2. with U and V independent. Then, MGF # Note: Prove F-distribution. Note: Use the proof of t-distribution as reference. X-½-F₁) % 3. Let X be a random sample from N(u,02). Find MLEs for and o². 4. Let X N(u,02). Find the MSE of the (uncorrected) sample variance -152 as an estimator for ² and compare with the MSE of the (corrected) sample variance $². 5. Let X Po(A). Find the CRLB for variances of unbiased estimators of T(X) = X.
Transcribed Image Text:1. Theorem: Let X = (X₁, X2, ..., X₁)' be a r.s. from N(ud), where μER and >0 and n ≥2. Then, a. X and S² are independent; 2. Defn: A continuous r.v. X is said to have an F-distribution with m (numerator) and n (denominator) degrees of freedom (d.f.) if, and only if, the PDF of X is given by Notation: Mean # Variance: T() rr) (1+x)(x), m.nez £x(x) = (m) ² X-F) E(X) = -²²2: "> n>2 Var(X) = DNE 2n² (m+n-2) m(n-2)(n-4) n>4 Theorem: Let U-X and V-2. with U and V independent. Then, MGF # Note: Prove F-distribution. Note: Use the proof of t-distribution as reference. X-½-F₁) % 3. Let X be a random sample from N(u,02). Find MLEs for and o². 4. Let X N(u,02). Find the MSE of the (uncorrected) sample variance -152 as an estimator for ² and compare with the MSE of the (corrected) sample variance $². 5. Let X Po(A). Find the CRLB for variances of unbiased estimators of T(X) = X.
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