**The Problem:** Let $X\thicksim\text{Binomial}(n,p)$ and $Y\thicksim\text{Bernoulli}(p)$ be independent random variables. Find the distribution of $X+Y.$

**My Attempt:** Set $Z=X+Y$. Then we can compute the distribution of $Z$ using the convolution formula because $X$ and $Y$ are independent. Also, since $X$ only takes values in $\{0,1,\dots,n\}$
and $Y$ only takes values in $\{0,1\}$, we have that $Z$ can only take values in $\{0,1,\dots,n+1\}$. Altogether, it follows that the probability mass function of $Z$ for $k\in\{0,1,\dots,n+1\}$ is
given by
\begin{equation*}\begin{split}
p_Z(k)&=\sum_{\ell\in\mathbb Z}p_X(k-\ell)p_Y(\ell)=p_X(k)p_Y(0)+p_X(k-1)p_Y(1)\\
&=\binom{n}{k}p^k(1-p)^{n-k+1}+\binom{n}{k}p^{k+1}(1-p)^{n-k}.
\end{split}\end{equation*}

**My Concern:** My issue with my work above is that we cannot plug in $n+1$ into $p_Z(k)$, which leads me to think that the range of $Z$ is $\{0,1,\dots,n\}$ instead. Could anyone please shed some light into this issue I am encountering? Also any comments on the rest of my attempt are very welcomed. Thank you very much for your time.