Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$

Multiple Choice

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a normal population with mean $\mu _ { 1 } \text { and known variance } \sigma _ { 1 } ^ { 2 } \text {, }$ and let $Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample from a normal population with mean $\mu _ { 2 } \text { and variance } \sigma _ { 2 } ^ { 2 } \text {, }$ and that the X and Y samples are independent of one another. Which of the following statements are true?

A) $\bar { X }$
Is normally distributed with expected value $\mu _ { 1 } \text { and variance }σ _ { 1 } ^ { 2 } / m$
B) $\bar { Y }$
Is normally distributed with expected value $\mu_{2} \text { and variance } σ_{2}^{2} / n$

C) $\bar { X } - \bar { Y }$
Is normally distributed with expected value $\mu _ { 1 } - \mu _ { 2 } \text { and variance } \left(σ _ { 1 } ^ { 2 } / m + \sigma _ { 2 } ^ { 2 } / n \right) \text {. }$
D) $\bar { X } - \bar { Y }$
Is an unbiased estimator of $\mu _ { 1 } - \mu _ { 2 }$
)
E) All of the above statements are true.