Question 4
Let $(X_i)$ a sample from a $N(\mu_x, \sigma_x^2)$ distribution, let $(Y_j)$ an independent sample from a $N(\mu_y, \sigma_y^2)$ distribution, and let
$\bar{X} = \frac{1}{14} \sum_{i=1}^{14} X_i$, $\bar{Y} = \frac{1}{7} \sum_{j=1}^{7} Y_j$,
$S_x^2 = \frac{1}{13} \sum_{i=1}^{14} (X_i - \bar{X})^2$, $S_y^2 = \frac{1}{6} \sum_{j=1}^{7} (Y_j - \bar{Y})^2$,
$\hat{\sigma}_x^2 = \frac{1}{14} \sum_{i=1}^{14} (X_i - \mu_x)^2$, $\hat{\sigma}_y^2 = \frac{1}{7} \sum_{j=1}^{7} (Y_j - \mu_x)^2$.
(a) Find $E(S_x^2)$ and $E(\hat{\sigma}_y^2)$.
(b) Find $E\left(\frac{S_y^2}{\hat{\sigma}_x^2}\right)$.
(c) Find $E\left(\frac{\hat{\sigma}_y^2}{S_x^2}\right)$.
