= Logarithmic distribution
{title2=$\mathbb P(X=k)=-\theta^k/(k\log(1-\theta))$}
= Log-series distribution
{synonym}
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 $\sum_{k\geq1}\theta^k/k=-\log(1-\theta)$ 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.
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