0.(2).A sample of n elements is to be tested. If Y is the lifespan of the first of the nelements to fail, it can be shown that the pdf of Y isg(2) = , exp ()y > 0.(a) Use integration to show...

Extracted text: A mass-production line manufactures electrical heating elements with lifespans X; which have independent exponential distributions with pdf 1 exp T; > 0. (2). 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 g(2) = , exp () y > 0. (a) Use integration to show that the mean lifespan X; is 0, and the variance is 0. (b) Show that the sample mean X = 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? 2.