Question

Problem 1 Let X 1 , X 2 , ⋯,Xn be a random sample from a normal distribution with mean µ and variance σ 2 . Consider e X as an estimator of e µ where X is the sample mean. Show that e is consistent estimator of e µ X .

Answer 1

$E(estimator)=E(\frac{1}{n}\displaystyle\sum_{i=1}^nX)$

$E(\bar{X})=\frac{1}{n}\sum E(xi)$

$E(\bar{X})=\frac{1}{n}\displaystyle\sum_{i=1}^nu$

$=E(\bar{X})=\frac{1}{n}.nu$

$E(\bar{X})=u$