Logarithmic distribution ($\mathbb{P}(X=k)=-{\theta }^k/(k\log (1-\theta ))$)

For $0<\theta <1$, this probability distribution on positive integers has mass $\mathbb{P}(X=k)={\theta }^k/(k[-\log (1-\theta )])$. The Taylor identity ${\sum }_{k\ge 1}{\theta }^k/k=-\log (1-\theta )$ proves normalization. Its expectation is $-\theta /((1-\theta )\log (1-\theta ))$, by summing the geometric series after multiplying the mass by $k$.

For an independent sample, the maximum-likelihood estimator solves the equation making this model mean equal to the sample mean. The model mean is strictly increasing: set $t=-\log (1-\theta )$ to obtain $(e^t-1)/t$, whose derivative is positive for $t>0$. Hence a sample mean greater than one determines a unique interior estimate. Pooling several categories changes the likelihood and generally changes this estimate; the ungrouped sample mean is no longer recoverable from the pooled category counts alone.