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Assignment 3 - Stat 5101(Geyer Fall 2009 Homework...

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Stat 5101 (Geyer) Fall 2009 Homework Assignment 3 Due Wednesday, September 30, 2009 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 3-1. Suppose that pr is a PMF on a sample space Ω, suppose X and Y are random variables in this probability model. Prove the following statements. (a) E ( X + Y ) = E ( X ) + E ( Y ). (b) If X ( ω ) 0 for all ω Ω, then E ( X ) 0. (c) If Y ( ω ) = a for all ω Ω, then E ( XY ) = aE ( X ). (d) If Y ( ω ) = 1 for all ω Ω, then E ( Y ) = 1. Do not use the axioms (these are the axioms). The problem is to prove that these statements follow from our earlier definition of PMF and expectation. 3-2. Suppose X has the uniform distribution on the set { 1 , 2 , 3 , 4 } , and suppose Y = X 2 . (a) Calculate E ( X ). (b) Calculate E ( Y ). (c) Calculate E ( Y/X ). (d) Calculate E ( Y ) /E ( X ). (e) Compare your answers in (c) and (d). Are they the same? Should they be the same?
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