Give an example to show that if not assuming independence of

[squenshl]

Give an example to show that if not assuming independence of X₁, X₂, ..., X_n it is possible to show that Var(1/n * sum from k = 1 to n of X_k) >> $$\sigma^2/n$$

 

[g_edgar]

What do you get if $X_1 = X_2 = ...$, which is sort of the EXTREME example of non-independence?

 

[squenshl]

Since X₁ = X₂ = ... = X_n.
This implies that Var((1/n)*nX₁) = n²$$\sigma^2$$/n² = $$\sigma^2$$ >> $$\sigma^2/n$$