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hw08 - random variable where itself is random and has...

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ORIE 360/560 Fall 2007 Assignment 8 Due Tuesday, Oct. 30 at 2:00 pm Problem 1. (a) Suppose ( U, V ) is uniformly distributed on the triangle with vertices (0 , 0) , (1 , 0) , (1 , 1). Find E [ V | U ] and E [ V ]. (b) Suppose X is uniform on (0 , 1) and, conditional on X , Y is uniform on (0 , X ). Then the random vector ( X, Y ) is distributed on the same triangle described part (a), although in this case the joint distribution is not uniform. Find E [ Y | X ] and E [ Y ]. (c) Compare E [ V ] from part (a) with E [ Y ] from part (b). Describe in a sentence or two why these differ in the way that they do. Problem 2. A lightbulb has a 5% chance of having a defective filament and a 95% chance of having a good filament. If the filament is good, the lifetime of the bulb is exponential with mean 600 hours. If the filament is bad, the lifetime is exponential with mean 50 hours. Find the expected lifetime of such a lightbulb. Problem 3. The frequency of accidents on a given highway fluctuates as a function of the weather. Suppose we choose to model the number of accidents on a given day as a Poisson(
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Unformatted text preview: λ ) random variable, where λ itself is random and has the Γ(2 ,κ ) distribution. (Here, κ is a constant.) Hence, higher values of λ 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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