A mass-production line manufactures electrical heating elements with lifespans X; which have independent exponential distributions with pdf exp I; > 0. A sample of n elements is to be tested. If Y is the lifespan of the first of the n elements to fail, it can be shown that the pdf of Y is s0) - () e exp y > 0. (a) Use integration to show that the mean lifespan X; is 0, and the variance is 6. (b) Show that the sample mean X = D X; is an unbiased estimator of 0. (c) By noting the similarity between f(x;) and g(y) or otherwise, deduce the mean and variance of Y. (d) Find the constant k such that kY is an unbiased estimator of 0. Is the estimator consistent? (e) Which of the two estimators, X or kY, would you prefer?

A mass-production line manufactures electrical heating elements with lifespans X; which have independent exponential distributions with pdf exp I; > 0. A sample of n elements is to be tested. If Y is the lifespan of the first of the n elements to fail, it can be shown that the pdf of Y is s0) - () e exp y > 0. (a) Use integration to show that the mean lifespan X; is 0, and the variance is 6. (b) Show that the sample mean X = D X; is an unbiased estimator of 0. (c) By noting the similarity between f(x;) and g(y) or otherwise, deduce the mean and variance of Y. (d) Find the constant k such that kY is an unbiased estimator of 0. Is the estimator consistent? (e) Which of the two estimators, X or kY, would you prefer?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 9T

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