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Unformatted text preview: PSTAT 120C: Assignment #7 May 29, 2008 These problems are related to the material on Bayesian inference from week 9 and 10. These are for practice and exam preparation and will not be graded. The solutions will be posted Wednesday, June 3. 1. Suppose that Y ∼ N ( μ,σ 2 ) and Z ∼ N ( ν,τ 2 ) are independent, and then X = Y + Z . (a) We create a new random variable W = τ σ Y- σ τ Z. Show that Cov( X,W ) = 0. ( For normal random variables a zero covariance implies independence. ) (b) Show that Y = σ 2 σ 2 + τ 2 X + στ σ 2 + τ 2 W. (c) If X and W are independent, then argue that E ( W | X ) = τ σ μ- σ τ ν. (d) Use the previous results to show that E ( Y | X ) = σ 2 σ 2 + τ 2 ( X- ν ) + τ 2 σ 2 + τ 2 μ. 2. Exercise 16.8 on page 807 in the textbook. 3. Two friends Bob and Carol decide to play 5 sets of tennis. Before they begin playing, Carol and Bob agree to a wager: he’ll pay her $40 dollars if she wins and she’ll pay him $60 if he wins (she’s a better tennis player). Carol wins two of the first three sets.wins (she’s a better tennis player)....
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This note was uploaded on 10/02/2009 for the course PSTAT 5E taught by Professor Eduardomontoya during the Spring '08 term at UCSB.
- Spring '08