# s1 - Stat 5102 (Geyer) 2003 Midterm 1 Problem 1 (a) The...

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Stat 5102 (Geyer) 2003 Midterm 1 Problem 1 (a) The likelihood is L ( θ ) = n Y i =1 θe - θx i = θ n n Y i =1 e - θx i = θ n exp ˆ - θ n X i =1 x i ! = θ n e - θn ¯ x n Then likelihood times prior is L ( θ ) g ( θ ) θ n e - θn ¯ x n θ α - 1 e - βθ = θ n + α - 1 e - ( n ¯ x n + β ) θ which is proportional to a Gamma( n + α,n ¯ x n + β ) density, so that is the pos- terior. Alternatively, one could just apply Theorem 6.3.4 in DeGroot and Schervish. (b) Say the posterior is Gamma( α 0 0 ) with α 0 = n + α and β 0 = n ¯ x n + β . The mean of this distribution is α 0 β 0 = n + α n ¯ x n + β Problem 2 This is just like part (a) of question 1 except that this isn’t a theorem in DeGroot and Schervish, and you have to go from the names of the distributions to their densities on your own. The probability density function for the data given the parameter is f ( x | λ ) = λ 3 Γ(3) x 2 e - λx , 0 < x < , 0 < λ < . and the prior probability density function for the parameter is

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## This note was uploaded on 10/28/2010 for the course STAT 5101 taught by Professor Staff during the Spring '02 term at Minnesota.

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s1 - Stat 5102 (Geyer) 2003 Midterm 1 Problem 1 (a) The...

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