X and Y are two independent uniform random variables with ranges [0,α] and [0,β], respectively (β>α).
How can I calculate the pdf of Z, when Z is the sum these two independent random variables (i.e., Z=X+Y)?
2026-04-03 14:51:25.1775227885
Density of sum of two independent uniform variables with different intervals
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Comment (outline and visualization): For $\alpha = 5,\,\beta = 10$ here is a histogram (blue bars) of a large random sample from the distribution of $Z,$ showing @kimchilover's trapezoid. The joint distribution of $X$ and $Y$ is the uniform distribution over a rectangle with density $f_{X,Y}(x,y).$ When you work out the limits on the integral (split into three parts) in @spaceisdarkgreen's Comment, you will get the PDF of $Z$ (red line).