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Homework Help: Easy Change of Variables

  1. Feb 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Given the equation
    [tex] U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex]
    find an expression for [itex] \hat U(\hat x) [/itex] given that change of variables
    [tex] x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn [/tex]
    and [itex] \hat U [/itex] is the U under this variable transformation.

    3. The attempt at a solution
    Using the fact that [itex] x= \frac n2 + \sqrt n \mu [/itex] it is easy to re-arrange to find that

    [tex] \mu^2 = \frac1n \left(x-\frac n2\right)^2 = \frac{x^2}n - x + \frac n4 [/itex]

    dividing by n, we get

    [tex] \frac{\mu^2}n = \hat x^2 - \hat x + \frac14 = \left( \hat x - \frac12 \right)^2 [/itex]

    Now I substitute this back into [itex] U(\mu) [/itex] to get

    [tex] \hat U(\hat x) = \frac2{\sqrt\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right] [/tex]

    The problem is that the solution is supposed to be

    [tex] \hat U(\hat x) = 2 \sqrt{\frac n\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right] [/tex]

    I can't seem to deduce where the factor of [itex] \sqrt n [/itex] comes up.
  2. jcsd
  3. Feb 18, 2010 #2
    Nobody? Nothing?
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