# h3 - Statistics 5101 Fall 2010 Homework Assignment 3 Solve...

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Statistics 5101, Fall 2010: Homework Assignment 3 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 sup- pose 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? 3-3. Suppose E ( X ) = 3 and E ( Y ) = 4. Calculate E (5 X + Y ). 3-4. Suppose X is a random variable having PMF given by x 1 2 3 4 5 f ( x ) 1 / 9 2 / 9 3 / 9 2 / 9 1 / 9 (a) Calculate E ( X ). (b) Calculate var( X ). 3-5. Suppose X is a Ber( p ) random variable and Y = 2 X - 1.

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• Fall '02
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• Statistics, Probability theory, mean vector, Calculate, Calculate Var, variance matrix

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