Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 7 / a

The central limit theorem states that if $W_1,W_2,\dots$ are independent and identically distributed random variables with finite mean $\mu$ and finite positive variance ${\sigma }^2$, thenHere the arrow means convergence in distribution: for every real $z$, the probability that the expression on the left is at most $z$ tends to $\Phi (z)$, the standard normal distribution function. If ${\sigma }^2=0$, the variables equal $\mu$ almost surely and their sample mean is exactly $\mu$.

Assuming $g$ is measurable, applying it separately to independent and identically distributed random variables preserves those properties. Thus $W_i=g(X_i)$ have common mean ${\mu }_g$ and variance ${\sigma }_g^2$. For ${\sigma }_g^2>0$, the central limit theorem givesHence, for large $n$, the approximate probability distribution isThe second argument is the variance. If ${\sigma }_g^2=0$, this sample mean equals ${\mu }_g$ almost surely, so the appropriate description is a point mass at ${\mu }_g$.