I am doing a game theory question where $X$ is uniform over $[0,100]$ and $Y$ is uniform over $[10,110]$, where $X$ and $Y$ are the respective valuations of two people . I want to find the density function for $Z=X+Y$ in order to calculate some probabilities.
I've seen many solutions where X and Y follow a uniform distribution over $[0,1]$ but I don't seem to be able to replicate it using this range.
I have drawn a diagram and calculated some probabilities, e.g. $Pr(X+Y>60)=7/8$ but I would like to check them and calculate harder probabilities using the convolution of the two densities. Any help would be greatly appreciated!
Write $X=100A,\,Y=10+100B,\,C=A+B$ so $Z=10+100C$. Then $C$ has an $n=2$ Irwin–Hall distribution, which can be obtained as per @zugzug's comment. In terms of Iverson brackets$$f_C(c)=|1-c|[c\in[0,\,2]],\,f_Z(z)=\frac{1}{100}|1-(z-10)/100|[z\in[10,\,210]].$$