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Mean of random vector

  1. Feb 7, 2014 #1
    1. The problem statement, all variables and given/known data

    What is the expected value of ||Ax+n|| where || || is the L2 norm and x and n are uncorrelated and E[n] = 0


    3. The attempt at a solution
    E[ norm of Y] = E[(Ax+n)' (Ax+n)] = E[(x'A'+n')(Ax+n)] = E[x'A'xA +x'T'n +n'Ax +n'n]
    the three last terms = 0 due to uncorrelatedness so = E[x'A'xA] = E[tr(x'A'xA)] = E[tr(x'xA'A)] = A'AE[x'x] does this reduce any further?
     
  2. jcsd
  3. Feb 7, 2014 #2

    haruspex

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    You mean E[x'A'Ax +x'A'n +n'Ax +n'n] , right?
     
  4. Feb 7, 2014 #3
    Yes. But the result is still the same since it is a scalar and we can take the trace of it to rearrange. So does A'AE[x'x] simplify any further?
     
  5. Feb 7, 2014 #4

    haruspex

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    x'A'Ax is a scalar; x'A'x is a scalar; (x'A'x)A is an n x n matrix; x'(A'xA) is meaningless (since xA is meaningless).
     
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