Give an example to show that if not assuming independence of

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This discussion illustrates the implications of not assuming independence among random variables X1, X2, ..., Xn. It provides a specific example where all variables are equal (X1 = X2 = ... = Xn), leading to a variance calculation that demonstrates Var((1/n) * sum from k = 1 to n of Xk) being significantly greater than σ²/n. The conclusion drawn is that under extreme non-independence, the variance can escalate, contradicting the assumption of independence.

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Give an example to show that if not assuming independence of X1, X2, ..., Xn it is possible to show that Var(1/n * sum from k = 1 to n of Xk) >> \sigma^2/n
 
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What do you get if X_1 = X_2 = ..., which is sort of the EXTREME example of non-independence?
 


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