Convolution between two random variables Given $X$ is a random variable which distributed uniformly in $[1,3]$
and $Y$ distributed uniformly in $[0,a]$.  I need to calculate the value of the density function of:
$$Z=X+Y,$$
at a point $b$ (i.e., calculate $f_Z(b)$).  I have some difficulties with the splitting of the integral with the bounds of $a$, $x$ and $b$.

Note: I don't know the bounds of $a$ and $b$ (these are not given).

Graphical comment. Let $a = 3.$ A histogram of 100,000 simulated realizations of $Z$ suggests the shape of the density function of $Z.$
set.seed(921)
x = runif(10^5, 1, 3);  y = runif(10^5, 0, 3)
z = x + y
summary(z)

      Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
     1.009   2.735   3.498   3.500   4.268   5.992 

hist(z, prob=T, br=30, col="skyblue2", 
     main="Simulated Dist'n of Z")


