In: Statistics and Probability
Show that ?̅ is an unbiased estimator for ?. ( please show all work completely)
We have to show that E () =
Now,
=
Xi / n
, i = 1 , 2 , 3 , ...
So,
E( ) = E(
Xi / n )
= 1 / n * E( Xi )
= 1 / n * [ E( X1 + X2 + X3 + ... + Xn) ]
= 1/n * [ E(X1) + E(X2) + E(X3) + ... + E(Xn) ]
Since X1 , X2 , X3 , ... Xn are random variables and expected
value for each is equal which is ,
= 1/n * ( +
+
+ ....n
times)
= 1 / n ( n)
=
Therefore, is an
unbiased estimator of
.