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Given $X_1,...,X_n$ are i.i.d. RVs drawn from $\mathcal{N}(0,1)$, and constants $c_1,...,c_n$.

Let $X = \sum_{i=1}^n X_i^2, Y = \sum_{i=1}^n (X_i-c_i)^2$, therefore $X$ and $Y$ are central and non-central chi-squared variables, respectively.

If chi-squared random variables $Z$ and $W$ are independent, the ratio $\frac{Z}{W}$ is a non-central F-distribution. However, in this case, can $\frac{Y}{X}$ be derived, or at least be approximated?

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