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Homework set 5, Stat 4158, Spring 2021 Due date: Wednesday, Feb. 17, 2021 Instructions: Answer all questions and write clearly and formally in complete sentences. This is a team homework. Each team should submit one set of answers, by one student. Write team number and names of all participating team members. All students should answer all questions on his/her own and present those to the team for discussion. 1. Let Y1, . . . , Yn be a random sample from a uniform distribution with pdf f(y|θ) = 12θ+1 , if 0 < y="" ≤="" 2θ="" +="" 1,0,="" otherwise.="" where="" θ=""> 0 is an unknown parameter. (a) Derive Eθ(Y ) and V (Y ), the mean and variance of the population distribution. (b) Derive a method of moments estimator (θ̂MoM) of θ, based on the sample Y1, . . . , Yn. (c) Derive the bias and MSE formulas for your estimator in part(b). (d) Derive the maximum likelihood estimator (θ̂MLE) of θ. (e) Derive the bias and MSE formulas for your MLE in part(d). [Hint: Find the distribution of Y(n) = max{Y1, . . . , Yn} using the cdf approach, then find the mean and variance of Y(n) etc.] 2. Suppose the lifetime Y of a certain type of electronic component follows a gamma distribution with pdf f(y|θ) = 1 θ2 ye−y/θ, y > 0, where θ > 0 is an unknown parameter. It follows that E(Y ) = 2θ and V (Y ) = 2θ2. Suppose n components are tested independently to obtain information about the life- time distribution. Let Y1, . . . , Yn denote the lifetimes of the n tested components. (a) Derive the maximum likelihood estimator (θ̂MLE) of θ, based on the sample Y1, . . . , Yn. (b) Derive the bias and MSE of the MLE derived in part(a). (c) Derive a method of moments estimator of θ and compare it with the MLE. (d) Suppose we tested 5 components and observed their lifetimes as 5080, 6250, 4800, 7200 and 6700, in hours. Calculate the maximum likelihood estimates of the mean and the variance of the lifetime distribution.