Transformation of r.v. with floor function

Question

$$X\in \text{Exp}(1),\quad Y\in [X]$$ Find the distribution of $Y$. This is what I have done ($f(x)$ is the distribution of $X$) $$P(Y=k)=P([X]=k)=P(k\leq X< (k+1))=\int_{k}^{k+1} f(x) dx=\int_{k}^{k+1}e^{-x}dx =\Big[-e^{-x}\Big]_{k}^{k+1}=-e^{-(k+1)}-(-e^{-k})=e^{-k}(1-e^{-1})$$ $$Y\in Ge(1-e^{-1})$$ But now I have learned that the answer should be $Y\in Ge(1-e^{\frac{-1}{a}})$. What is correct?