(Solved): (6 pt) Suppose \( X_{1}, \ldots, X_{m} \) are a random sample from \( N\left(\mu_{1}, \sigma_{1}^{ ...

(6 pt) Suppose \( X_{1}, \ldots, X_{m} \) are a random sample from \( N\left(\mu_{1}, \sigma_{1}^{2}\right) \) and \( Y_{1}, \ldots, Y_{n} \) are a random sample from \( N\left(\mu_{2}, \sigma_{2}^{2}\right) \), independent from the \( X \) 's. Let \[ \bar{X}=\frac{1}{m} \sum_{j=1}^{m} X_{j} \quad \text { and } \quad \bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i} \] Give the probability distributions (just their names and parameters) of the following: \[ \bar{X}, \bar{Y}, \bar{X}+\bar{Y}, 2 \bar{X}-3 \bar{Y}, 4 \bar{X}+3 \bar{Y}, \text { and } \frac{\left(X_{1}-\mu_{1}\right)^{2}}{\sigma_{1}^{2}} \]