Let X1, .., Xn be a random sample from a population with 0 unknown and given by the density = * itz>0 { e f(x; 8) if x < 0 202 and E(X) = 0 (Hint: you may use that e-² za-1dz = Show that E(X) (a – 1)! for every a E N). 1. 2. Show that the statistic (1) n i=1 is an unbiased estimator of 0. 3. Give the definition of a consistent estimator. 4. Show that the estimator 0n given in relation (1) is a consistent estimator of 0. 5. Show that the estimator 0, is a minimum variance estimator of 0. (Hint: use the Cramer-Rao inequality given by 1 var(0) > (Ə ln(f(X;0) nE

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A: 0 \)" data-mce-style="cursor: default;"&gt;f(x)=c(1+x)4;x&gt;0

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