We estimate the unknown mean θ of a random variable X (where X has a finite and positive variance) by forming the sample mean Mn=(X1+⋯+Xn)/n of n i.i.d. samples Xi and then forming the estimator

Θˆ=Mn+1n.

Is this estimator unbiased?

unanswered
Is this estimator consistent?

unanswered
Consider now a different estimator, Θˆn=X1 , which ignores all but the first measurement.

Is this estimator unbiased?

unanswered
Is this estimator consistent?