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Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function 0xe-0 x! 0, a) b) c) f(x; 0) = Show that for any & > 0 and Sn = Σ=₁ X₁, lim P(|Sn − 0| ≥ ɛ) = 0. 12-00 Let 8₁= Show that a statistic Så in a) is the maximum likelihood estimator of the parameter 0. X₁+2X₂+2X3-X4 = for x = 0,1,2,... elsewhere and Ô₂ = ²(X₁ + X₂ + X3 + X4) be two unbiased estimators of 0. Which one of the two estimators is more efficient?

Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function 0xe-0 x! 0, a) b) c) f(x; 0) = Show that for any & > 0 and Sn = Σ=₁ X₁, lim P(|Sn − 0| ≥ ɛ) = 0. 12-00 Let 8₁= Show that a statistic Så in a) is the maximum likelihood estimator of the parameter 0. X₁+2X₂+2X3-X4 = for x = 0,1,2,... elsewhere and Ô₂ = ²(X₁ + X₂ + X3 + X4) be two unbiased estimators of 0. Which one of the two estimators is more efficient?

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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Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function a) b) c) d) f(x; 0) = Let ₁ 0xe-0 x! = 0, } Show that for any &> 0 and S₂ = ₁X₁, lim P(|S₂ − 0| ≥ ɛ) = 0. n→∞0 for x = 0,1,2,... elsewhere Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0. X₁+2X₂+2X3-X4 4 and Ô₂ = ²¹ (X₁ + X₂ + X3 + X4) be two unbiased estimators of 0. Which one of the two estimators is more efficient? What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter 8? e) Use the one-parameter regular exponential family definition to find the functions, h(x), c(0), w (0) and t(x).
Transcribed Image Text:Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function a) b) c) d) f(x; 0) = Let ₁ 0xe-0 x! = 0, } Show that for any &> 0 and S₂ = ₁X₁, lim P(|S₂ − 0| ≥ ɛ) = 0. n→∞0 for x = 0,1,2,... elsewhere Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0. X₁+2X₂+2X3-X4 4 and Ô₂ = ²¹ (X₁ + X₂ + X3 + X4) be two unbiased estimators of 0. Which one of the two estimators is more efficient? What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter 8? e) Use the one-parameter regular exponential family definition to find the functions, h(x), c(0), w (0) and t(x).
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