i dont know how to do it

Question

The three independent observations X1 , X2 , and , have a normal distribution
N (µ , σ = 1) with an unknown expectation µ . Show that the following two estimates of
µ are unbiased: ˆµ1 = X1 , µˆ2 = X1 − X2 + X3 .

Answer 1

Unbiased means E[estimate]=true parameter. So try to calculate

E[ˆµ1]=µ. I think it is not hard to do that.

Indeed, E[ˆµ1]=E[X1]=∫xφ(x)dx=µ, where φ(x) is the normal density with mean µ and variance 1. The only thing you have to is to evaluate

∫xφ(x)dx. 

As for X1-X2+X3 is very similar. E[X1-X2+X3]=E[X1]-E[X2]+E[X3]=µ.