$\symSetX$, respectively. Analogously, the sketch for $\symSetS_2$ could have been obtained from sketches for $\symSetB$ and $\symSetX$. Let $\symRegValVariate_\symSetASuffix$, $\symRegValVariate_\symSetBSuffix$, and $\symRegValVariate_\symSetXSuffix$ be the values of the considered register in the sketches for $\symSetA$, $\symSetB$, and $\symSetX$, respectively. The corresponding values in sketches for $\symSetS_1$ and $\symSetS_2$ are given by $\symRegValVariate_1 = \max(\symRegValVariate_\symSetASuffix, \symRegValVariate_\symSetXSuffix)$ and $\symRegValVariate_2 = \max(\symRegValVariate_\symSetBSuffix, \symRegValVariate_\symSetXSuffix)$. Since $\symSetA$, $\symSetB$, and $\symSetX$ are disjoint and therefore $\symRegValVariate_\symSetASuffix$, $\symRegValVariate_\symSetBSuffix$, and $\symRegValVariate_\symSetXSuffix$ are independent, the joint cumulative probability function of $\symRegValVariate_1$ and $\symRegValVariate_2$ is $$\begin{gathered}
\label{equ:joint_cdf}
\symProbability(
\symRegValVariate_1 \leq \symRegVal_1
\wedge
\symRegValVariate_2 \leq \symRegVal_2
)
=
\\
\symProbability(
\symRegValVariate_\symSetASuffix \leq \symRegVal_1)
\,
\symProbability(
\symRegValVariate_\symSetBSuffix \leq \symRegVal_2)
\,
\symProbability(
\symRegValVariate_\symSetXSuffix \leq \min(\symRegVal_1, \symRegVal_2)
)
=
\\
\begin{cases}
0 & \symRegVal_1 < 0 \vee \symRegVal_2 < 0
\\
e^{
-
\frac{\symPoissonRate_{\symSetASuffix}}{\symNumReg 2^{\symRegVal_1}}
-
\frac{\symPoissonRate_{\symSetBSuffix}}{\symNumReg 2^{\symRegVal_2}}
-
\frac{\symPoissonRate_{\symSetXSuffix}}{\symNumReg 2^{\min(\symRegVal_1, \symRegVal_2)}}
}
& 0\leq\symRegVal_1 \leq \symRegRange \wedge 0\leq\symRegVal_2\leq\symRegRange
\\
e^{
-
\frac{\symPoissonRate_{\symSetBSuffix} + \symPoissonRate_{\symSetXSuffix}}{\symNumReg 2^{\symRegVal_2}}
}
& 0\leq\symRegVal_2 \leq \symRegRange < \symRegVal_1
\\
e^{
-
\frac{\symPoissonRate_{\symSetASuffix} + \symPoissonRate_{\symSetXSuffix}}{\symNumReg 2^{\symRegVal_1}}
}
&  0\leq\symRegVal_1 \leq \symRegRange < \symRegVal_2
\\
1
&
\symRegRange < \symRegVal_1 \wedge \symRegRange < \symRegVal_2.
\end{cases}\end{gathered}$$ Here we used that $\symRegValVariate_\symSetASuffix$, $\symRegValVariate_\symSetBSuffix$, and $\symRegValVariate_\symSetXSuffix$ follow under the Poisson model. The corresponding probability mass 