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Question 6 22 pts Consider an i.i.d. sample X1, .... X, from the continuous probability distribution D(0) with density function f(x) = V21 03 where o e (0, co) is an unknown parameter. (a) Verify that f(x) as defined above is a valid probability density function. [Hint: it may help to recall some properties of the standard normal distribution.] 2 pts (b) Explain why EX, = 0, regardless of the value of 0. 2 pts (c) Show that Var X, = 30". [Hint: if Z ~ N(0, 1), then it is known (e.g., from the moment-generating function Be* =e"/2) that EZ' = 3.] 3 pts (d) Find the method-of-moments estimator OMM of 8. 2 pts (e) Show that is an unbiased estimator of 0 , and find its mean-squared error. 4 pts (f) Find the maximum-likelihood estimator o of o given the sample X1. .. .. X- Also, evaluate o for a specific sample with sample size n = 50, sample mean X = -0.71 and sample variance s = 62.13. 5 pts (g) Suppose that n is very large. Using the large-sample properties of maximum-likelihood estimators, calculate the asymptotic standard deviation of the maximum-likelihood estimate o in terms of n and d. 2 pts (h) Hence give a formula for an approximate 95% confidence interval for o in terms of n and the observations X1. ..., X,, and evaluate this to give a numerical 95% confidence interval for o in the case of the specific sample observed in (f). 2 pts