Quiz 1, Stat 4158, Spring 2021 Time: 20 minutes. Instructions: This quiz is open book and notes. Explain your answers clearly. Scan and upload your work as one pdf file. 1. Let Y be a random variable...

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Quiz 1, Stat 4158, Spring 2021 Time: 20 minutes. Instructions: This quiz is open book and notes. Explain your answers clearly. Scan and upload your work as one pdf file. 1. Let Y be a random variable with pdf f(y|α, θ) = αx α−1 θα , 0 < x="">< θ,="" α=""> 0, θ > 0. (a) Show that the expected value and variance of Y are E(Y ) = ( α α + 1 )θ and V (Y ) = αθ2 (α + 2)(α + 1)2 . (b) Suppose Y1, . . . , Yn is a random sample from the above distribution and that α is known (i.e., given). (i) Derive a method of moment estimator of θ. (ii) Derive the bias and variance of the estimator you derive in part (i).