Sum of uniformly distributed random variables in a given range

Question

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given by Irwin-Hall distribution. But what if the random variables are uniformly distributed over the interval [0-W]?

Answer 1

If we replace uniforms on $(0,1)$ by uniforms on $(0,w)$, the resulting random variable $Y$ has the same distribution as $wX$, where $X$ has Irwin-Hall distribution. In particular, $\Pr(Y\le y)=\Pr(wX\le y)=\Pr(X\le \frac{y}{w})$. It follows that if $f_X$ is the density function of the Irwin-Hall, then $Y$ has density $f_Y(y)=\frac{1}{w}f_X(y/w)$.

In a similar way, one can read off almost any desired facts about the distribution of $Y$ from related facts about the Irwin-Hall.