= Solution
{c}
\b[The printed numerical example reverses the digits and contradicts both definitions.] With the stated last-digit definition, $[Z]=17$ gives $X=7$, $Y=1$. Use those definitions throughout.
Put $q=e^{-\lambda}\in(0,1)$ and $N=\lfloor Z\rfloor$. The <exponential distribution> gives
$$
\mathbb P(N=n)=\mathbb P(n\le Z<n+1)=(1-q)q^n,\qquad n\ge0.
$$
The <Euclidean division> $N=10Y+X$ has unique remainder $X\in\{0,\ldots,9\}$ and quotient $Y\in\{0,1,\ldots\}$. Thus the <joint probability mass function> is
$$
\boxed{p_{X,Y}(x,y)=(1-q)q^{x+10y},\quad
x=0,\ldots,9,\ y=0,1,\ldots,}
$$
and zero otherwise. This is <decimal digit decomposition of a geometric random variable>. Following the erroneous example instead would require different definitions and would not answer the stated question.
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