I am studying for actuary exam 1/p and I have questions about bivariate probability notation. When I'm solving I am solving double integrals with a nested f(x,y)= xy + y^2 I notice that the problems ask for Pr (X<.5) or Pr(Y < .35). In math theory, how does this output coincide to the inputs? As far as I can tell, the difference is the way the integration is supposed to be ordered, starting with either dy or dx, and placing an x or y as a limit on the first integral.
Please help!
Normally, $$ \mathbb{P}\left[(X,Y) \in \Omega \subseteq \mathbb{R}^2\right] = \iint_A f_{X,Y}(x,y) dA $$ so for example, $$ \mathbb{P}[X<x] = \int_{-\infty}^x \int_{-\infty}^\infty f_{X,Y}(x,y) dydx = \int_{-\infty}^\infty \int_{-\infty}^x f_{X,Y}(x,y) dxdy $$ and similarly you can compute $\mathbb{P}[Y < y]$ by integrating the $x$ out over $(-\infty,\infty)$.