Question

Problem 4. The standard error of an estimator is defined as the standard deviaition of that estimator. In class we introduced the sample mean X,-(1/n) Σί ix, as an estinator of E(X] where Xi's are iid samples of the random variable X. What is the standard error of the estimator Xn? Assume that the statndard deviation of X is σ.

Answer 1

Here one needs to keep in mind that $X_{i}$ 's are identically independently distributed random variables.

Standard Deviation =$\sqrt{Variance}$

hence in this Standard Deviation(X)=$\sigma$

hence Varience(X)=$\sigma ^{2}$

Standard deviation is denoted as S.D

Standard Error is denoted as S.E

variance=V(X)

As,

$X_{i}$ 's are identically independently distributed random variables.

V($X_{i}$ )=$\sigma ^{2}$  $\forall$ i=1,2,....n

and Cov($X_{i},X_{j}$)=0   $\forall$ i$\neq$j

also V(aX)=$a^{2}$ V(X) where a is any real constant