= Integer and fractional parts of an exponential variable
{title2=$I=\lfloor Z\rfloor,\quad U=Z-I$}
For exponential rate $\lambda>0$, the integer part has mass $(1-e^{-\lambda})e^{-\lambda m}$ on nonnegative integers. The fractional part has density $\lambda e^{-\lambda u}/(1-e^{-\lambda})$ for $0\leq u<1$. The mixed joint density factors into those marginals, proving <independence>. The integer part is a <geometric distribution> on zero-based support, with <expectation> $1/(e^\lambda-1)$.
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