midterm3practice - Practice problems for Final Exam Problem...

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Practice problems for Final Exam Problem 1 A sample X 1 , X 2 , . . . , X n comes from a continuous dis- tribution with the density f X ( x ) = ( θ + 1) x θ , 0 < x < 1 for some unknown parameter θ > - 1. ( a ) Compute the maximum likelihood estimator of θ . ( b ) Compute the moment estimator of θ . ( c ) Compute the Bayes estimator of θ that maximizes the posterior density if the prior density of θ is p ( θ ) = e - ( θ +1) for θ > - 1. Problem 2 Suppose that the daily maximal air temperatures X 1 and X 2 , observed on two successive days follow a bivariate normal distribution with parameters μ 1 = μ 2 = 75, σ 1 = σ 2 = 8 and ρ = . 9. ( a ) Find the probability that the average over two successive days of the daily maximal air temperatures exceeds 80. ( b ) Find the probability that the maximal air temperature on the second day exceeds 80 given that the maximal air temperature on the first day was equal to 80. Problem 3 Suppose that X has the uniform distribution in the interval [0 , T ], but T itself is random, and can take values 1 and 2 with probabilities 1 / 2 each. ( a ) Compute the mean and the variance of X . ( b ) Find the probability P ( X > 1). Problem 4 A sample A sample X 1 , . . . , X n comes from a continuous distribution with the density f X ( x ) = θ ( θ - 1) x (1 + x ) θ +1 , x > 0 for some unknown parameter θ > 1.
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