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h2 - Stat 5101(Geyer Fall 2011 Homework Assignment 2 Due...

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Stat 5101 (Geyer) Fall 2011 Homework Assignment 2 Due Wednesday, September 21, 2011 Solve each problem. Explain your reasoning. No credit for answers with no explanation. 2-1. Suppose we have a PMF f X with domain S (the original sample space), and we have a map g : S T that induces a probability model with PMF f Y with domain T (the new sample space) given by the formula on slide 82. Prove that for any real-valued function h on T X y T h ( y ) f Y ( y ) = X x S h ( g ( x ) ) f X ( x ) 2-2. Suppose we have the uniform distribution on S = {- 2 , - 1 , 0 , 1 , 2 } and the random variable X defined by X ( s ) = s 2 , s S. Determine the PMF of the random variable X . 2-3. Suppose X = ( X 1 , X 2 ) has the uniform distribution on { 1 , 2 , 3 , 4 , 5 , 6 } 2 . (a) Show that the components of X are independent. (b) Determine the PMF of the distribution of the random variable Y = X 1 + X 2 . (c) Determine E ( Y ). (d) Define μ = E ( Y ). Determine E { ( Y - μ ) 2 } . 2-4. Suppose X = ( X 1 , X 2 , X 3 ) has the uniform distribution on the set { (0 , 0 , 0) , (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } Show that the components of X
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