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Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively.

Then show that

$\frac{S_1^2}{S_2^2} $ follows $F$-distribution with $\left(n_1-1\right)$ and $\left(n_1-1\right)$ degrees of freedom.

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HINT:

Note that $$\dfrac{S_1^2}{\sigma^2}\sim \chi^2_{\displaystyle n_1-1} \text{and} \dfrac{S_2^2}{\sigma^2}\sim \chi^2_{\displaystyle n_2-1}$$.

Hence we get $$\displaystyle \dfrac{S_1^2}{\sigma^2} / \dfrac{S_2^2}{\sigma^2} =\dfrac{S_1^2}{S_2^2}\sim F_{\displaystyle n_1-1,n_2-1}$$

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