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5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...

5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ.

6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1.

(a) Show that g(y|α) is a pdf when c = 3 6 + 2α .

(b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be used to calculate a MOM estimator of α. Subsequently, use M2 to construct a MOM estimator of α based on a random sample y1, . . . , yn.

(c) Construct the log-likelihood function for the parameter α. Show that the ML estimator of α can be found by solving the equation 1 3 + α = 1 n Xn j=1 y 2 j 1 + αy2 j .

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