I am having trouble finding the integral bounds when I compute the sum of two independent, continuous random variables. For example:
Let $X$ be an exponential variable with parameter $\lambda$ and $Y$ be a uniform random variable on $[0,1]$ independent of $X$. Find the probability density function of $X+Y$.
So, we have that $f_X(x)=\lambda e^{-\lambda x}$ and $f_Y(y)=\begin{cases} 1, \hspace{0.2in} 0\leq y \leq 1 \\ 0 \hspace{0.2in} \text{ otherwise} \end{cases}$
The following formula applies here: $$f_{X+Y}(a)=\int_{-\infty}^\infty f_X(a-y)f_Y(y)dy=\int_{-\infty}^\infty f_Y(a-x)f_X(x)dx$$
Then, $$f_{X+Y}(a)=\int_0^a \lambda e^{-\lambda (a-y)}dy=1-e^{-\lambda a}$$
However, the answer is the following: $$f_{X+Y}(a)=\begin{cases} 1-e^{-\lambda a}, \hspace{0.2in} 0 \leq a \leq 1 \\ e^{-\lambda a}(e^{\lambda}-1) ,\hspace{0.2in} a \geq 1 \\ 0 \hspace{0.2in} \text{otherwise} \end{cases}$$
It seems that my process only found the solution for the interval $0\leq a \leq 1$. How would I find the solution for the interval $a \geq 1$? My textbook is rather vague about finding these integral boundaries, and I was wondering if there is a good method for this, that I can apply to varying combinations of distributions $X$ and $Y$. Thank you.