Question

Let X 1 , X 2 , ⋯,Xn be a random sample from a normal distribution with mean µ 

 and variance σ2. Consider e X as an estimator of e µ where X  is the sample  

µ mean. Show that e X is consistent estimator of e .

Answer 1

Showing that X is a consistent ( unbiased ) estimator for u.

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

$=\frac{1}{n}\sum E(x)$

=$\frac{1}{n}\sum u$

$\frac{1}{n} n.u$

Therefore $E( estimator )= u$

For clarity on the missing information in the equations see the picture attached.