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B2. Let A₁,..., An> 0 and let X₁,..., Xn be independent random variables with common mean E(X) = μ and variances Var (X₂) = 1/X₁. Let a₁,..., an ER be given. Consider the estimator (X₁,..., Xn) for the mean , given by n μ(21,...,2n) = Σαρα i=1 (f) Assume that is unbiased. Using the Cauchy-Schwarz inequality, or otherwise, show that the variance Var((X₁,..., Xn)) is minimal, if a₁ = λί/Σ=1 λ; for all i € {1,...,n}. (Hint: consider the sum Σi=1;√√₁.)

B2. Let A₁,..., An> 0 and let X₁,..., Xn be independent random variables with common mean E(X) = μ and variances Var (X₂) = 1/X₁. Let a₁,..., an ER be given. Consider the estimator (X₁,..., Xn) for the mean , given by n μ(21,...,2n) = Σαρα i=1 (f) Assume that is unbiased. Using the Cauchy-Schwarz inequality, or otherwise, show that the variance Var((X₁,..., Xn)) is minimal, if a₁ = λί/Σ=1 λ; for all i € {1,...,n}. (Hint: consider the sum Σi=1;√√₁.)

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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B2. Let A₁,..., An> 0 and let X₁,..., Xn be independent random variables with common mean E(X₂) = μ and variances Var(X;) = 1/λ₁. Let α₁,..., an E R be given. Consider the estimator (X₁,..., Xn) for the meanu, given by n μ(21,...,Xn) = Σαΐti i=1 (f) Assume that is unbiased. Using the Cauchy-Schwarz inequality, or otherwise, show that the variance Var ((X₁,..., Xn)) is minimal, if α₁ = ₁/₁=1 Aj for all i € {1,..., n}. (Hint: consider the sum Σi=1₁√₁₂)
Transcribed Image Text:B2. Let A₁,..., An> 0 and let X₁,..., Xn be independent random variables with common mean E(X₂) = μ and variances Var(X;) = 1/λ₁. Let α₁,..., an E R be given. Consider the estimator (X₁,..., Xn) for the meanu, given by n μ(21,...,Xn) = Σαΐti i=1 (f) Assume that is unbiased. Using the Cauchy-Schwarz inequality, or otherwise, show that the variance Var ((X₁,..., Xn)) is minimal, if α₁ = ₁/₁=1 Aj for all i € {1,..., n}. (Hint: consider the sum Σi=1₁√₁₂)
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