Find the density function of the random variable $Z=X+Y$ Let $X$ and $Y$ be independent and uniformly distributed random variable on the intervals $[0,3]$ and $[0,1]$, respectively. Find the density function of the random variable $Z=X+Y$.
I find $f(x,y)=1/3$ and the range of $Z$ to be $[0,4]$, but I cannot find the density function of $Z$. 

Another way, if you are confortable with convolutions, is to recall that the sum of independent variables has a density that is the convolution of the components. In this case, we must convolve a unit rectangle with a rectangle in $[0,3]$, which is very similar to this 
image (except for some axis shifting and scaling) (in our case we'd have $f_X$ in red, $f_Y$ in blue, and the result in green), and which results in the function given in Calvin's answer.

