= Fisher information per observation and per trial
{c}
{title2=$I_{\mathrm{geom}}(\theta)=I_{\mathrm{Bern}}(\theta)/\theta$}
Precision comparisons using <Fisher information> must specify what one observation records. A <Bernoulli distribution> with success probability $\theta$ gives information $I_{\mathrm{Bern}}=1/[\theta(1-\theta)]$ per trial. A <geometric distribution> on $1,2,\ldots$ records the number of trials until the first success. Its log-likelihood for one waiting time $X$ is $\log\theta+(X-1)\log(1-\theta)$, so
$$
I_{\mathrm{geom}}=\frac1{\theta^2}+\frac{\mathbb E(X-1)}{(1-\theta)^2}
=\frac1{\theta^2(1-\theta)}.
$$
At equal numbers of observations, geometric waiting times therefore provide $1/\theta$ times as much information, and the asymptotic <variance> of their <maximum-likelihood estimator> is $\theta$ times the Bernoulli variance. But each waiting time uses an expected $1/\theta$ trials. Information per expected underlying trial is consequently $\theta I_{\mathrm{geom}}=I_{\mathrm{Bern}}$. This separates an observation count from the cost of collecting the observations.
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