An Exercise of noncentral [math]\chi^2[\math] distribution.

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An Exercise of noncentral chi-square distribution.

Let \(Y_1,\ldots,Y_n\) be independent random variables with \(Y_k\) distributed as \(N\sim(a_k,\sigma^2) \), and \(\bar Y=\sum_{k=1}^{n}\frac{Y_k}{n}\) denote the sample mean, \(S^2\) denotes the sample variance.

Then show that
\(\frac{(n-1)S^2}{\sigma^2}=\frac{1}{\sigma^2}\sum_{k=1}^{n}(Y_k-\bar Y)^2\)

is distributed as noncentral \(\chi^2\) with \( (n-1) \) degrees of freedom and noncentrality parameter \(\lambda=\sum_{k=1}^n\frac{(a_k-\bar a)^2}{\sigma^2}\), where \(\bar a=\sum_{k=1}^{n}\frac{a_k}{n}\).

I have only the idea of Moment Generating Function(MGF) Technique to prove this type of proof based on transformation. But this is almost impossible to me use the MGF technique for the given exercise.

How can i prove it easily?
 
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