 Solution

For the same sample size $n$, the geometric distribution provides more Fisher information:The corresponding asymptotic variances of the maximum-likelihood estimators areHence choose the geometric sample for more precise inference at equal numbers of observations. Its asymptotic variance is $\theta$ times as large, and its standard error is $\sqrt{\theta }$ times as large. This illustrates Fisher information per observation and per trial. It is a per-observation comparison: one geometric observation records a waiting time containing, on average, $1/\theta$ Bernoulli trials. A constraint on the total number of underlying trials would be a different sampling-budget comparison.