Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$

Multiple Choice

Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$ with X and Y independent variables, and let $\hat { p } _ { 1 } = X / m \text { and } \hat { p } _ { 2 } = Y/ n$ Which of the following statements are not correct?

A) $E \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } - p _ { 2 } \text {, so } \hat { p } _ { 1 } - \hat { p } _ { 2 }$
Is an unbiased estimator of $p _ { 1 } - p _ { 2 }$
B) When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { \mathbf { 2 } }$
Individually has approximately normal distributions.
C) When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { 2 }$
Has approximately a normal distribution.
D) $V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n , \text { where } q _ { l } = 1 - p _ { l } \text { for } i = 1,2$
E) All of the above statements are correct.