assign6_2015.pdf - MAST30027 Modern Applied Statistics...

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MAST30027: Modern Applied Statistics Assignment 6 Due: 1:00 pm Fri 23 October (week 12) This assignment is worth 3 1/3% of your total mark. Here is a histogram of 299 observations of the time between eruptions for the Old Faithful geyser in Yellowstone National Park. The data can be found in the file geyserdata.txt on the subject website. We will model this data using a mixture of two normals, which has density f ( x ) = πσ - 1 1 φ (( x - μ 1 ) 1 ) + (1 - π ) σ - 1 2 φ (( x - μ 2 ) 2 ) where φ is the standard normal density, π [0 , 1], μ i R and σ i (0 , ). (a) Show that if A bin(1 , π ), X 1 N ( μ 1 , σ 2 1 ) and X 2 N ( μ 2 , σ 2 2 ), all independent of each other, then Y = AX 1 + (1 - A ) X 2 has density f . (b) To build a Bayesian model for Y ( i ), the i -th observation, we introduce variables G ( i ) such that if G ( i ) = j then Y ( i ) has mean μ j and precision τ j = 1 2 j , for j = 1 , 2. Moreover P ( G ( i ) = 1) = π and P ( G ( i ) = 2) = 1 - π . Thus, to specify the model we need priors for π , μ j and τ j . To avoid ambiguity, we suppose that
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