I'm trying to generate random sample with distribution, that can get by sum of independent $\xi$=Exp(0.5) and $\eta$=Unif(-1, 1), and compare empirical cdf and theoretical cdf.
I'm using R to do that
z <- rexp(500, 0.5) + runif(500, -1, 1)
Then I want to check result and make plot of theoretical cdf, but I've stuck on convolution. I know that
$$f_{\xi+\eta}(z)=\int_{-\infty}^{+\infty} f_\xi(x)⋅f_\eta(z−x)dx$$
in my case
$$f_{\xi+\eta}(z)=\int_{-\infty}^{+\infty} 0.5*e^{-0.5x} 1\{0 \le x\} * 1/2 * 1\{-1\le z-x\le1\} dx$$
and then
$$f_{\xi+\eta}(z)=\int_{0}^{+\infty} 0.25*e^{-0.5x} * 1\{x-1\le z\le x+1\} dx$$
What should I do on the next steps?
Hint:
The support of the distribution $G$ of an exponential distributed random variable $T$ and an $-1,1$-uniform random variable $U$, $T$ and $U$ independent, is $(-1,\infty)$. $G$ has density given by given by $$g(x)=\frac12\int_{\mathbb{R}}\lambda e^{-\lambda t}\mathbb{1}_{(0,\infty)}(t)\mathbb{1}_{(-1,1)}(x-t)\,dt=\frac{\lambda}{2}\int^\infty_0\mathbb{1}_{(x-1,x+1)}(t)e^{-\lambda t}\,dt$$
If $x\leq-1$, then $g(x)=0$.
If $-1<x\leq1$, $$g(x)=\frac{\lambda}{2}\int^{x+1}_0 e^{-\lambda t}\,dt$$
if $1<x$, then $$g(x)=\frac{\lambda}{2}\int^{x+1}_{x-1}e^{-\lambda t}\,dt$$