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5102-Midterm-1-Practice-Problems

# 5102-Midterm-1-Practice-Problems - PRACTICE PROBLEMS FOR...

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PRACTICE PROBLEMS FOR MIDTERM 1 STAT 5102-004 (1) What is a statistic? (2) Prediction intervals are wider than confidence intervals. Why? (3) A Bernoulli Y with mean θ has variance θ (1 - θ ). How large a random sample guarantees that E | ¯ Y - θ | 2 0 . 01? Note that 0 a 1 implies 0 a (1 - a ) 1 / 4. (4) Let Y 1 , . . . , Y n iid N ( μ, 1). What n guarantees Pr( | ¯ Y n - μ | ≤ 0 . 049) 0 . 95. (5) Let Y 1 , . . . , Y n iid N (0 , σ 2 ) with σ 2 unknown. Let τ = 1 σ 2 . The density of each random variable is (2 π ) - 1 2 τ 1 2 exp ( - 1 2 τy 2 ) . (a) Confirm that ˆ τ = n n i =1 Y 2 i is the mle for τ . (b) Find the mle ˆ σ 2 for σ 2 . (c) Confirm that n ˆ σ 2 σ 2 χ 2 ( n ). (6) Let Y Gamma( α, β ). Note that Y is positive with density, mean, and variance f ( y | α, β ) = β α Γ( α ) x α - 1 e - βx E Y = α β Var Y = α β 2 . If α = 1, then Y is an exponential random variable. Let Y 1 , . . . , Y n iid exponential( θ ), and assume a gamma prior distribution for θ . (a) Show that the posterior is gamma, and specify the parameters.

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5102-Midterm-1-Practice-Problems - PRACTICE PROBLEMS FOR...

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