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Gaussian variables

  1. Dec 28, 2011 #1
    I am not sure how to do the following hw question:

    Suppose X1, X2, ... are independent Gaussian variables with mean zero and variance 1. Consider the event that

    X1 + X2 + .... + X2n ≥ 2na wth a > 0

    Compare the chance of observing this event in the following two ways:
    (i) by getting that X1 + X2 + ... + Xn ≥ na and Xn+1 + Xn+2 + ... +X2n ≥ 2na

    (ii) by getting that X1 + X2 + .... + Xn ≥ 2na and Xn+1 + Xn+2 + .... + X2n ≥ 0

    I tried letting Y1 = X1 + ... + Xn and Y2 = Xn+1 + ... + X2n.

    For (i), Y1 and Y2 are each normally distributed with mean 0 and variance n,
    so we have P(Y1 > an)P(Y2 > an) = P(Y1 > an)^2.

    For (ii), P(Y1 > 2an)(1/2).
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 29, 2011 #2

    Ray Vickson

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    Science Advisor
    Homework Helper


    What is the problem? Both methods give wrong answers, just different wrong answers.

    RGV
     
  4. Dec 30, 2011 #3
    I need to know which approach is closer to the answer. They are both special cases of the actual event
     
  5. Dec 30, 2011 #4
    The integrals u get for (i) and (ii) are hard to compare
     
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