Decimal digit decomposition of a geometric random variable ($N=bY+X$)

The same argument works for any integer base $b\ge 2$. Suppose $N$ has geometric distribution $\mathbb{P}(N=n)=(1-q)q^n$, $n\ge 0$, with $0<q<1$. Its Euclidean division $N=bY+X$ gives $0\le X<b$ and $Y\ge 0$. Their joint probability mass function isThe factors are normalized probability mass functions, so the final digit $X$ and quotient $Y$ are independent. The quotient again has geometric distribution, with success probability $1-q^b$. If $N$ is the integer part of an exponential random variable of rate $\lambda$, take $q=e^{-\lambda }$.