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Hamilton, on page 101 of his textbook Statistics in Physical Science says that the non-central $\chi^2$ distributed random variable with $k$ degrees of freedom can be though of as the sum of squares of $k$ independent normal random variables $x_i \sim \mathcal{N}(\mu_i,\sigma_i^2), i = 1,...,k$.

Many other sources say that the i.i.d. normal r.v.s must all have unit variance. How do I derive the non-central $\chi^2$ distribution from $\mathcal{N}(\mu_i,\sigma_i^2)$ r.v.s?

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