For an exponential distribution with mean 0, a random sample of size 7 is drawn. The...

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For an exponential distribution with mean 0, a random sample of size 7 is drawn. The 25th smoothed empirical percentile is X(2), the 2nd smallest value of the sample. 6x 0 (a) Show that the density of X(2) is fx(2)(x) 42 = -(e 0 (b) Hence, express E[X(2)] and Var[X(2)] in terms of 0. 7x -e) for x>0. [Hint: You may use the following fact without proof: ^x"e*dx=- .] m! km+1 Let the population 25th percentile be t. (c) Express in terms of 0. (d) is estimated by matching X(2) to 7. Show that this results in the estimator X(2) In(4/3) Find the mean square error of this estimator of in terms of 0. For an exponential distribution with mean 0, a random sample of size 7 is drawn. The 25th smoothed empirical percentile is X(2), the 2nd smallest value of the sample. 6x 0 (a) Show that the density of X(2) is fx(2)(x) 42 = -(e 0 (b) Hence, express E[X(2)] and Var[X(2)] in terms of 0. 7x -e) for x>0. [Hint: You may use the following fact without proof: ^x"e*dx=- .] m! km+1 Let the population 25th percentile be t. (c) Express in terms of 0. (d) is estimated by matching X(2) to 7. Show that this results in the estimator X(2) In(4/3) Find the mean square error of this estimator of in terms of 0.