Convolution of uniform distributions

Question

If $X$ is a random variable with a continuous uniform distribution on (0,10), and $Y$ is also from a continuous uniform distribution on (0,10) how do I calculate the CDF of $X+Y$?

Answer 1

You can compute this directly as follows. First, for any $0<t<20$ write

$$ F(t):=\mathbb{P}(X+Y\leq t) = \int_{0}^{10} \mathbb{P}(X+y\leq t) \frac{1}{10}\, dy $$

Case I: $20>t\geq 10$, then

$$ F(t)= \int_{0}^{10} \frac{\min{\{t-y,10}\}}{100}\, dy = 1-\frac{(20-t)^2}{100}. $$

Case II: $0<t<10$, then $$ F(t)= \int_{0}^{10} \frac{\max{\{t-y,0}\}}{100}\, dy = \frac{t^2}{200}. $$