I've seen the Irwin-Hall distribution which is for the sum of n independent variables with a standard uniform distribution.
However, is there a more general version that would help me find the distribution of W = X + Y - Z , where X, Y and Z are independent and are each uniformly distributed from 0 to alpha?
Many thanks for any tips.
Let's call $IH_3$ an Irwin–Hall distribution for the sum of $3$ independent and identically distributed $U(0, 1)$ random variables, so with support on $[0,3]$ with mean $\frac32$ and variance $\frac14$
If you are looking at $X+Y+Z$ the sum of $3$ independent and identically distributed $U(0, \alpha)$ random variables, then this distribution would be a rescaling by $\alpha$ of $IH_3$, so with support on $[0,3\alpha]$ with mean $\frac32\alpha$ and variance $\frac14\alpha^2$
Since $Z$ and $\alpha-Z$ have the same distribution and are each independent of $X$ and $Y$, if you were looking at $X+Y+(\alpha-Z)$, then this distribution would also be a rescaling by $\alpha$ of $IH_3$, so again with support on $[0,3\alpha]$ with mean $\frac32\alpha$ and variance $\frac14\alpha^2$
If you are looking at $X+Y-Z$, then this distribution would be a rescaling by $\alpha$ of $IH_3$ then location-shifted by $-\alpha$, so with support on $[-\alpha,2\alpha]$ with mean $\frac12\alpha$ and variance $\frac14\alpha^2$