I could use some help with this - thanks!

Suppose X₁, ... , X_n are i.i.d. normally distributed with mean _X and variance ². Let X_n be the sample mean. Consider estimating . First, consider the case where the mean is known. For any given constant a, define the estimator

^n,a=nai=1nXi

(i) Compute E(^n,a )

(ii) For what a is ^n,a a consistent estimator of ? Explain.

(iii) Now consider the case where is unknown, which means that ^n,a cannot be computed. Instead, for any given constant c, define the estimator:

^n,c=nci=1nXiXn

Compute E(^n,c )

(iv) For what c is ^n,c a consistent estimator of ? Explain.