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2010100AHW4

2010100AHW4 - h X in terms of long run averages(5 Calculate...

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STAT 100A HWIV Problem 1: Suppose we roll a biased die, the probability mass function is p (1) = . 1, p (2) = . 1, p (3) = . 1, p (4) = . 2, p (5) = . 2, and p (6) = . 3. Let X be the random number we get by rolling this die. (1) Calculate P ( X > 4). Calculate P ( X = 6 | X > 4). (2) Calculate E( X ). (3) Suppose the rewards for the six numbers are respectively h (1) = - \$20, h (2) = - \$10, h (3) = \$0, h (4) = \$10, h (5) = \$20, and h (6) = \$100. Calculate E[ h ( X )]. (4) Please interpret E( X ) and E[
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Unformatted text preview: h ( X )] in terms of long run averages. (5) Calculate Var( X ). (6) Calculate Var[ h ( X )]. Problem 2: For a discrete random variables X , (1) Prove E[ aX + b ] = a E[ X ] + b . (2) Prove Var[ aX + b ] = a 2 Var[ X ]. (3) Let μ = E[ X ] and σ 2 = Var[ X ]. Let Z = ( X-μ ) /σ , calculate E[ Z ] and Var[ Z ]. (4) Prove Var[ X ] = E[ X 2 ]-E[ X ] 2 . 1...
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