How was this integral set up to compute $Pr(X+Y) \geq\frac{\pi}{2}$?

Question

I am trying to understand how to deal with the following type of question given two random variables $X$ and $Y$ that are jointly continuous with some pdf:

Here: $f_{X,Y}(x,y) = \left\{ \begin{align} x\cos y, 0 \leq x \leq \frac{\pi}{2},0 \leq y \leq x \\ 0 \text{ , otherwise} \end{align} \right.$

Now I don't know how to calculate $Pr(X+Y) \geq\frac{\pi}{2}$

I have the answer and they choose bounds for their integrals

$$\int_{\frac\pi 4}^{\frac\pi 2} \int_{\frac\pi 2 -x}^x \mathrm{d }y \mathrm{d }x$$

How were these bounds chosen? Why was this integral correct.

Thanks for your help.

Answer 1

Without the constraints of the pdf, how would you describe the region where $x+y \ge \pi/2$?

It would be something like $\int_{-\infty}^\infty \int_{\frac{\pi}{2}-x}^\infty \mathop{dy}\mathop{dx},$ the area above the line $y=\frac{\pi}{2}-x$.

If you put the pdf as the integrand, then the support (where the pdf is nonzero) narrows down the region further, to give you the integral you wrote.

Answer 2

For such problems, draw the region first, and carefully go over the bounds mentioned.

The limits of the integral should be clear from this.