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?

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.8: Fitting Exponential Models To Data

Problem 3TI: Table 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to...

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