(adsbygoogle = window.adsbygoogle || []).push({}); 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 issupposedto 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.

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# Homework Help: Easy Change of Variables

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