Past exam of the statistics course of the University of Oxford / 2017 / 1 / a / iii / Solution

For a Bernoulli distribution with fixed $0<p<1$, the mean and variance are $p$ and $p(1-p)$. The odds transformation $g(x)=x/(1-x)$ has derivative $g^{\prime }(x)=(1-x{)}^{-2}$. Applying the delta method from part (ii),More precisely, $\sqrt{n}\{\overline{X}/(1-\overline{X})-p/(1-p)\}$ converges to a normal distribution of mean zero and variance $p/(1-p{)}^3$, after defining the statistic arbitrarily on $\overline{X}=1$. That exceptional event has probability $p^n\to 0$, so this extension does not change the limit. The unextended ratio is undefined there and has no finite exact variance if interpreted as infinite. At $p=0$ it is zero almost surely; at $p=1$ the denominator is always zero. This is the delta method for Bernoulli odds.