Sum of Two (General) Uniform Random Variables

Question

I would like to know if there is a general formula for calculating the sum of two uniform distributions $X$ and $Y$, where $X$ is uniformly distributed on $(-a,a)$ and $Y$ is uniformly distributed on $(-b,b)$, where $a>b>0$.

Thanks a lot in advance!

Answer 1

Are $X$ and $Y$ independent? If so, what you want to use is the convolution. If $X$ and $Y$ are any independent variables with density functions $f_X$ and $f_Y$, respectively, then the density function $f_{X+Y}(z)$ for $X+Y$ is given by $$ f_{X+Y}(z)=\int_{-\infty}^{\infty}f_X(x)\cdot f_Y(z-x)\,dx. $$ (This makes sense, if you think about it; the event "$X+Y=z$" is the same as the event "there is an $x$ so that $X=x$ and $Y=z-x$".)

You need to use the specific density functions for your uniform variables to simplify and compute this integral. That will give you the density function, and whatever else you want can be determined from there!

If you run through this, it will enable you to come up with exactly the formula you want. Just make sure to think carefully about the different cases for $z$ in the integral.