 Integer and fractional parts of an exponential variable

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 $0\le u<1$. The mixed joint density factors into those marginals, proving independence. The integer part is a geometric distribution on zero-based support, with expectation $1/(e^{\lambda }-1)$.