A random variable \( X \) can take the values 0,1 and 2 with \( P(X=0)=1-3 \alpha / 4 \), \( P(X=1)=\alpha / 2 \) and \( P(X=2)=\alpha / 4 \) for some parameter \( \alpha>0 \). One observation is taken and we would like to estimate \( \alpha \).
i. Consider the estimators \( T_{1}=X \) and \( T_{2}=2 X^{2} / 3 \). Show that they are both unbiased estimators of \( \alpha \).
ii. Which of these estimators do you prefer and why?