= Solution
{c}
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> $g'(x)=(1-x)^{-2}$. Applying the <delta method> from part (ii),
$$
\boxed{\frac{\overline X}{1-\overline X}\mathrel{\dot\sim}
N\!\left(\frac p{1-p},\frac{p}{n(1-p)^3}\right).}
$$
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> $p^n\to0$, 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>.
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