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Problem 5. let > 0 be an unknown parameter and suppose that X₁ and X₂ are independent random variables, each having the exponential distribu- tion with density function f(x) = ex — Since f(xlu) has mean it is clear that an unbiased estimator of u is X₁ + X₂ 2 a. Calculate the variance of the estimator X of . b. Now consider the estimator of u given by T(X₁, X₂)=√√√X₁X₂. Calculate the bias of T(X₁, X2), i.e., calculate E(T (X₁, X₂)) — . - T(X₁, X₂) is strictly smaller X x > 0. c. Show that the mean square error of T than the mean square error of X, where MSE(T) = E [(T(X₁, X2) -H)2], MSE(X) = E(X-μ)²]. =

Problem 5. let > 0 be an unknown parameter and suppose that X₁ and X₂ are independent random variables, each having the exponential distribu- tion with density function f(x) = ex — Since f(xlu) has mean it is clear that an unbiased estimator of u is X₁ + X₂ 2 a. Calculate the variance of the estimator X of . b. Now consider the estimator of u given by T(X₁, X₂)=√√√X₁X₂. Calculate the bias of T(X₁, X2), i.e., calculate E(T (X₁, X₂)) — . - T(X₁, X₂) is strictly smaller X x > 0. c. Show that the mean square error of T than the mean square error of X, where MSE(T) = E [(T(X₁, X2) -H)2], MSE(X) = E(X-μ)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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Problem 5. let > 0 be an unknown parameter and suppose that X₁ and X2 are independent random variables, each having the exponential distribu- tion with density function S(zl) == exp(-). f(x\μ) wwwww... x > 0. Since f(x) has mean it is clear that an unbiased estimator of u is X₁ + X₂ 2 X a. Calculate the variance of the estimator X of μ. b. Now consider the estimator of a given by T(X₁, X2) = √√X₁ X₂. Calculate the bias of T(X₁, X2), i.e., calculate E(T(X₁, X2)) - μ. c. Show that the mean square error of T = T(X₁, X2) is strictly smaller than the mean square error of X, where MSE(T) = E [(T(X₁, X₂) -μ)²], MSE(X)= E [(X -μ)²].
Transcribed Image Text:Problem 5. let > 0 be an unknown parameter and suppose that X₁ and X2 are independent random variables, each having the exponential distribu- tion with density function S(zl) == exp(-). f(x\μ) wwwww... x > 0. Since f(x) has mean it is clear that an unbiased estimator of u is X₁ + X₂ 2 X a. Calculate the variance of the estimator X of μ. b. Now consider the estimator of a given by T(X₁, X2) = √√X₁ X₂. Calculate the bias of T(X₁, X2), i.e., calculate E(T(X₁, X2)) - μ. c. Show that the mean square error of T = T(X₁, X2) is strictly smaller than the mean square error of X, where MSE(T) = E [(T(X₁, X₂) -μ)²], MSE(X)= E [(X -μ)²].
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