What is the variance of a mean of a set of n independent and identically distributed random variables?

V(xbar) | X ~ iid

Question

What is the variance of a mean of a set of n independent and identically distributed random variables?

V(xbar) | X ~ iid

BlackTom AI · Accepted Answer

The question asks for the variance of the mean (sample average) of a set of n independent and identically distributed random variables.

Key concept: If X1, X2, ..., Xn are IID with variance Var(X) = sigma^2, then the sample mean X̄ = (1/n) ∑ Xi has variance Var(X̄) = Var( (1/n) ∑ Xi ).

Because the Xi are independent, variances add: Var(∑ Xi) = ∑ Var(Xi) = n sigma^2. Scaling by 1/n yields Var(X̄) = (1/n^2) * n sigma^2 = sigma^2 / n.

Restating in the notation used: Var(X̄) = V(X) / n, where V(X) denotes the variance of the underlying IID random variable X.

Since the provided answer option is V(X)/n, this aligns with the standard result for the variance of the sample mean of IID variables.

Note on the options: The prompt shows an answer list with a single item and the answer options array appears empty. If other options existed, they would need to be evaluated against the same derivation, but with the current information the correct textbook result is Var(X̄) = Var(X)/n.

In practice, this means that as n grows larger, the dispersion of the sample mean decreases proportionally to 1/n, reflecting the averaging effect reducing variability.

What is the variance of a mean of a set of n independent and identically distributed random variables? V(xbar) | X ~ iid单项选择题

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