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Unformatted text preview: ) random variable, where itself is random and has the (2 , ) distribution. (Here, is a constant.) Hence, higher values of correspond to weather more conducive to creating lots of accidents. Find the expected number of accidents on a randomly selected day. Problem 4. Find the expected value of X conditional on the event X > 1 given that X is (a) Uniform on (0,2). (b) Exponential with mean 1. (c) Poisson with mean 1. Problem 5. Prove the variance decomposition formula , Var[ X ] = Var[ E [ X | Y ]] + E [Var[ X | Y ]] . The following problems are optional for students enrolled in 360, required for students enrolled in 560. Problem 6. Prove that if X and Y are independent random variables, then E [ X | Y ] = E [ X ]. (You may assume X and Y are continuous if you like.) Problem 7. Prove that for any random variables X and Y , E [ XY | Y ] = Y E [ X | Y ]. (Again, you may assume X and Y are continuous.)...
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- Fall '07