Hello I have already one problem that stipulates:
Consider two independent random samples $\mathsf{X_1,X_2,\ldots,X_n}$ and $\mathsf{Y_1,Y_2,\ldots,Y_m}$ from the respective normal distribution $N(\mu_x,\sigma^2_x)$ and $N(\mu_y,\sigma^2_y)$ where the four parameters are unknown.
Given $0<\alpha<1$, the problem asks to find some expressions for the random variables $L$ and $U$ such that $$P\left(L<\frac{\sigma^2_x}{\sigma^2_y}<U\right) = 1-\alpha$$
If someone might check on this my work please:
I have that $$ \displaystyle\frac{\frac{\frac{(m-1)S_y^2}{\sigma^2_y}}{(m-1)}}{\frac{\frac{(n-1)S_x^2}{\sigma^2_x}}{(n-1)}}= \frac{\sigma^2_x\,S^2_y}{\sigma^2_y\,S^2_x} \sim F(m-1,n-1)$$ has Fisher distribution with $m-1 \,\text{and}\,n-1$ degrees of freedom where $S^2_x\, \text{and}\,S^2_y$ are the respective sample variances.
So, $$P\left(L<\frac{\sigma^2_x}{\sigma^2_y}<U\right) = 1-\alpha$$
$$ P\left(L\frac{S^2_y}{S^2_x}<\frac{\sigma^2_x\,S^2_y}{\sigma^2_y\,S^2_x}<U\frac{S^2_y}{S^2_x}\right) = 1-\alpha$$
$$f_{1-\frac{\alpha}{2}}(m-1,n-1) = L\frac{S^2_y}{S^2_x}\quad \Longrightarrow L = \frac{S^2_x}{S^2_y}\frac{1}{f_{\frac{\alpha}{2}}(n-1,m-1)}$$
and $$f_{\frac{\alpha}{2}}(m-1,n-1) = U\frac{S^2_y}{S^2_x}\quad \Longrightarrow U = \frac{S^2_x}{S^2_y}\,{f_{\frac{\alpha}{2}}(m-1,n-1)}$$
Is this alright? or have I mistakes? please if somebody may help me thank you in advance.