Consider the following weighted averages of independent random variables X1, X2 , X3  all with mean µ and variance σ2 :  ˆθ1 = 1/3 X1 + 1/3 X2 + 1/3 X3    ˆθ2 = 1/4 X1 + 2/4 X2  + 1/4 X3  ˆθ3 = 2/5 X1  + 2/5 X2  + 2/5 X3  (a) Find E[ ˆθ1], E[ ˆθ2], and E[ ˆθ3]  (b) Are ˆθ1, ˆθ1, and ˆθ1 unbiased for µ? Please Explain.  (c) Find the variance for ˆθ1, ˆθ1, and ˆθ1 (d) If you had to use one of the above unbiased estimators, which would you pick? Explain.

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