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**Unformatted text preview: **Exercises Chapter 2 2.1 Marginal and conditional probability: The social mobility data from Sec- tion 2.5 gives a joint probability distribution on ( Y 1 ,Y 2 )= (father’s oc- cupation, son’s occupation). Using this joint distribution, calculate the following distributions: a) the marginal probability distribution of a father’s occupation; b) the marginal probability distribution of a son’s occupation; c) the conditional distribution of a son’s occupation, given that the father is a farmer; d) the conditional distribution of a father’s occupation, given that the son is a farmer. 2.2 Expectations and variances: Let Y 1 and Y 2 be two independent random variables, such that E[ Y i ] = μ i and Var[ Y i ] = σ 2 i . Using the definition of expectation and variance, compute the following quantities, where a 1 and a 2 are given constants: a) E[ a 1 Y 1 + a 2 Y 2 ] , Var[ a 1 Y 1 + a 2 Y 2 ]; b) E[ a 1 Y 1- a 2 Y 2 ] , Var[ a 1 Y 1- a 2 Y 2 ]. 2.3 Full conditionals: Let X,Y,Z be random variables with joint density (dis- crete or continuous) p ( x,y,z ) ∝ f ( x,z ) g ( y,z ) h ( z ). Show that a) p ( x | y,z ) ∝ f ( x,z ), i.e. p ( x | y,z ) is a function of x and z ; b) p ( y | x,z ) ∝ g ( y,z ), i.e. p ( y | x,z ) is a function of y and z ; c) X and Y are conditionally independent given Z . 2.4 Symbolic manipulation: Prove the following form of Bayes’ rule: Pr( H j | E ) = Pr( E | H j )Pr( H j ) ∑ K k =1 Pr( E | H k )Pr( H k ) where E is any event and { H 1 ,...,H K } form a partition. Prove this using only axioms P1- P3 from this chapter, by following steps a)-d) below: a) Show that Pr( H j | E )Pr( E ) = Pr( E | H j )Pr( H j ). P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 BM2, c Springer Science+Business Media, LLC 2009 226 Exercises b) Show that Pr( E ) = Pr( E ∩ H 1 ) + Pr( E ∩ {∪ K k =2 H k } ). c) Show that Pr( E ) = ∑ K k =1 Pr( E ∩ H k ). d) Put it all together to show Bayes’ rule, as described above. 2.5 Urns: Suppose urn H is filled with 40% green balls and 60% red balls, and urn T is filled with 60% green balls and 40% red balls. Someone will flip a coin and then select a ball from urn H or urn T depending on whether the coin lands heads or tails, respectively. Let X be 1 or 0 if the coin lands heads or tails, and let Y be 1 or 0 if the ball is green or red. a) Write out the joint distribution of X and Y in a table. b) Find E[ Y ]. What is the probability that the ball is green? c) Find Var[ Y | X = 0], Var[ Y | X = 1] and Var[ Y ]. Thinking of variance as measuring uncertainty, explain intuitively why one of these variances is larger than the others....

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