Which statement best describes the asymptotic distribution of the sample mean for i.i.d. variables with finite mean and variance?

Which statement best describes the asymptotic distribution of the sample mean for i.i.d. variables with finite mean and variance?

• A. It becomes exactly normal for any n.
• B. It converges to a Poisson distribution.
• C. It converges to a uniform distribution on an interval.
• D. It is approximately normal with mean mu and variance sigma^2/n for large n.

Answer: (D)

The key idea is the Central Limit Theorem. If you have independent, identically distributed observations with finite mean mu and finite variance sigma^2, the sampling distribution of the average Xbar becomes approximately Normal as the sample size n grows. Specifically, Xbar is distributed approximately Normal(mu, sigma^2/n) for large n, and equivalently sqrt(n)(Xbar - mu) tends to Normal(0, sigma^2). This means the variability of the mean shrinks like sigma/sqrt(n) and the distribution becomes bell-shaped regardless of the original distribution, as long as the variance is finite. So the statement describing the sample mean as approximately normal with mean mu and variance sigma^2/n for large n best captures the asymptotic behavior. The other options don’t fit: exact normality for any n only holds in special cases (like when the underlying distribution is already normal); a Poisson or a uniform distribution does not describe the general asymptotic distribution of the mean in this setting.

The key idea is the Central Limit Theorem. If you have independent, identically distributed observations with finite mean mu and finite variance sigma^2, the sampling distribution of the average Xbar becomes approximately Normal as the sample size n grows. Specifically, Xbar is distributed approximately Normal(mu, sigma^2/n) for large n, and equivalently sqrt(n)(Xbar - mu) tends to Normal(0, sigma^2). This means the variability of the mean shrinks like sigma/sqrt(n) and the distribution becomes bell-shaped regardless of the original distribution, as long as the variance is finite.

So the statement describing the sample mean as approximately normal with mean mu and variance sigma^2/n for large n best captures the asymptotic behavior. The other options don’t fit: exact normality for any n only holds in special cases (like when the underlying distribution is already normal); a Poisson or a uniform distribution does not describe the general asymptotic distribution of the mean in this setting.