Question

Assume n independent observations, denoted Xi, (i=1,....n), are taken from a distribution with a mean of E(X)=μ and variance V(X) =σ2. Prove that the mean of the n observations has an expected value of E(X)=μ and a variance of V(X) =σ2/n. Use the appropriate E and V rules in your answer. What happens as n becomes large? What does this tell you about the quality of the sample mean as an estimate of μ as the sample size increases?

Answer 1

We have,

                      

Consider,

         

                      

                      

                      

                      

                      

     Hence,

Similarly,

             

                                               

                        

                          

                          

                          

Hence,

  

We observe that, Variance of sample mean is inversely proportional to sample size. That mean when sample size is large then variance of mean is less. Therefore, When sample size large(n large) then the estimate of mean is good estimate.