Question

Question #9e57d

Answer 1

see below

Explanation:

if`" "barX` is the sample mean of a distribution and `mu` is the population mean then to prove that the sample mean is an unbiased estimator we need to show that

`E(barX)=mu`

let `X_1,X_2,X_3,...X_n" "`be a random sample of `n` independent values taken from any population (with replacement)then

`E(barX)=E(1/n(X_1+X_2+X_3+...+X_n))`

using the properties of expectation

`E(barX)=1/nE(X_1+X_2+X_3+...+X_n)`

`E(barX)=1/n[E(X_1)+E(X_2)+E(X_3)+...+E(X_n)]`

`:.E(barX)=1/n(mu+mu+mu+...+mu)`

`E(barX)=1/nxxn mu=mu`

as required