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can anyone help me transform a function of this form:

$$ \int f(x) \ e^{-(\frac{x-\mu}{\sigma})^2} \ dx $$

into a form that looks like this:

$$ \int g(y) \ e^{-y^2} \ dy $$

reason: I have to solve the first integral using Gauss–Hermite quadrature which can solve only the second type of integrals.

I have tried

$$ \int f(x*\sigma + \mu) \ e^{-x^2} \ dx $$

but it's not the same (trasnformed, correct). Can anyone tell me what I'm missing here.

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  • $\begingroup$ It should be dy in second integral $\endgroup$ – ABC May 5 '13 at 15:28
  • $\begingroup$ @exploringnet yes thank you. Fixed it. $\endgroup$ – zidarsk8 May 5 '13 at 15:44
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Put $$\dfrac{x-\mu}{\sigma}=y\ =>\dfrac{dx}\sigma=dy$$ and $${\sigma}{f(\sigma y +\mu)}=g(y)$$

I's sigma times whole expression you gave. The $dx=\sigma dy$ and not just $dy$

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