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t1 - Stat 5102 First Midterm Exam Name Student ID 1 The...

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Stat 5102 First Midterm Exam February 26, 2003 Name Student ID The exam is open book and open handouts. You may also use one 8 1 2 × 11 sheet of paper with formulas, etc. You may use a calculator, but shouldn’t need to. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from. The points for the questions total to 100. There are 2 pages and 5 prob- lems. 1. [20 pts.] Suppose X 1 , X 2 , ... , X n are independent and identically distributed with probability density function f ( x | θ ) = θe - θx , 0 < x < , 0 < θ < , and suppose the prior distribution for θ has probability density function g ( θ ) = β α Γ( α ) θ α - 1 e - βθ , 0 < θ < , where α > 0 and β > 0 are known numbers (the hyperparameters). (a) Find the posterior distribution of θ given data x 1 , ... , x n . You may specify the posterior distribution either by giving its probability
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