= Delta method for Bernoulli odds
{title2=$\sqrt n(\widehat o-o)\xrightarrow dN(0,p/(1-p)^3)$}
For independent <Bernoulli random variables> of fixed success <probability> $0<p<1$, put $\widehat p=n^{-1}\sum_iX_i$ and $o=p/(1-p)$. The <central limit theorem> gives $\sqrt n(\widehat p-p)\xrightarrow dN(0,p(1-p))$. Applying the <delta method> to the <odds> map $g(p)=p/(1-p)$, whose <derivative> is $(1-p)^{-2}$, gives
$$
\sqrt n\left(\frac{\widehat p}{1-\widehat p}-\frac p{1-p}\right)\xrightarrow dN\!\left(0,\frac p{(1-p)^3}\right).
$$
The ratio is undefined on the all-success <event>, whose <probability> is $p^n$. Any finite extension there has the same limiting <probability distribution>, since that <probability> tends to zero. This is an asymptotic statement and does not assert an exact finite-sample <variance> of the unextended ratio. It is useful for uncertainty in estimated <odds> when the success <probability> stays away from one.
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