Calculate $P(|X-Y| \geq L/4)$ for two independent uniform random variable $X$ and $Y$

Question

$X$ is a uniform random variable on [$0$, $L/2$] and Y is an uniform random variable on [$L/2$,$L$]. $X$ and $Y$ are independent. Calculate $P(|X-Y| \geq L/4)$.

This is what I have so far.

since $X$ and $Y$ are independent, the joint probability density function of $X$ and $Y$ is:

$$f(x,y) = \begin{cases} \frac{4}{L^2} &\textrm{ when } 0\leq x \leq \frac{L}{2},\frac{L}{2} \leq y \leq L, \\ 0&\textrm{ otherwise } . \end{cases}$$

and $P(|X-Y| \ge L/4)$ = $P(X-Y \leq -\frac{L}{4})$ + $P(X-Y \geq \frac{L}{4})$

I'm having trouble setting up the integration limit, can someone help me please?

Answer 1

When the density is constant and the event is a simple region bounded by linear equations, I find it easier to draw the regions and calculate the areas than to perform a formal integration. In this case the event corresponds to two triangles whose areas are readily calculated.