about the cardinality which makes the multiset $\lbrace\symRegValVariate_1,\ldots,\symRegValVariate_\symNumReg\rbrace$ a sufficient statistic for $\symCardinality$. Since the values of the multiset are all in the range $[0, \symRegRange+1]$, the multiset can also be written as $\lbrace\symRegValVariate_1,\ldots,\symRegValVariate_\symNumReg\rbrace = 0^{\symCountVariate_0}1^{\symCountVariate_1}\cdots\symRegRange^{\symCountVariate_{\symRegRange}}(\symRegRange+1)^{\symCountVariate_{\symRegRange+1}}$ where $\symCountVariate_\symRegVal$ is the multiplicity of value $\symRegVal$. As a consequence, the multiplicity vector $\boldsymbol{\symCountVariate} := (\symCountVariate_0,\ldots,\symCountVariate_{\symRegRange+1})$, which corresponds to the register value histogram, is also a sufficient statistic for the cardinality. By definition we have $\sum_{\symRegVal=0}^{\symRegRange+1}\symCountVariate_{\symRegVal}=\symNumReg$.

Poisson Approximation {#sec:poisson_approximation}
---------------------

The multinomial distribution is the reason that register values are statistically dependent and that further analysis is difficult. For simplification a Poisson model can be used [@Flajolet2007], which assumes that the cardinality itself is distributed according to a Poisson distribution $\symCardinality \sim \text{Poisson}(\symPoissonRate)$. Under the Poisson model the register values are independent and identically distributed according to $$\label{equ:register_value_distribution}
\symProbability(\symRegValVariate \leq \symRegVal\vert\symPoissonRate)
=
\begin{cases}
0 & \symRegVal < 0 \\
e^{-\frac{\symPoissonRate}{\symNumReg 2^\symRegVal}} & 0\leq \symRegVal \leq \symRegRange \\
1 & \symRegVal > \symRegRange.
\end{cases}$$

The Poisson approximation makes it easier to find an estimator $\symPoissonRateEstimate = \symPoissonRateEstimate(\boldsymbol{\symRegValVariate})$ for the Poisson rate $\symPoissonRate$ than for the cardinality $\symCardinality$ under the fixed-size model. Depoissonization finally allows to translate the estimates back to the fixed-size model. Assume 