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Question: 1 let y1 y be a random...

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1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so that and hence determine a 100(1-α)% confidence interval for θ (e) Suppose n 10 and to test Ho : θ 1 against H1 : θ 1.05 the null hypothesis is rejected if Y(n)2 0.99 i. What is the probability of committing a Type I error? ii. What is the probability of committing a Type II error? 13) 13]
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