Fisher information per observation and per trial ($I_{\mathrm{geom}}(\theta )=I_{\mathrm{Bern}}(\theta )/\theta$)

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 $1,2,\dots$ records the number of trials until the first success. Its log-likelihood for one waiting time $X$ is $\log \theta +(X-1)\log (1-\theta )$, soAt 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.