Say $X$ and $Y$ are 2 continuous random variables, where $f_X$ is the p.d.f of X, and $f_Y$ that of $Y$. I was wondering whether $\mathbb{P}[X>Y]$ is equivalent to the area between the curves $f_X$ and $f_Y$ in regions where the former function is greater than the latter.
I am conflicted on this, because while this does make intuitive sense on some level, I have been taught that the density function doesn't correspond to the exact probability at a certain place, but rather the 'probability per unit length'- and finding the area between the 2 curves seems to use this notion of the density function as giving the probability at $x$. So, is $\mathbb{P}[X>Y]$ the area between the curves $f_X$ and $f_Y$ in regions where the former function is greater than the latter?
No. Let $X \sim U(0,1)$ and $Y \sim U(1,2)$. Then $\mathbb{P}(X > Y) = 0$, but $f_X = 1_{(0,1)}$ and $f_Y = 1_{(1,2)}$, so the area between $f_X$ and $f_Y$ in regions where the former is greater than the latter is $1$. If you don't like that the supports of $X$ and $Y$ aren't equal, you could add a $N(0,\varepsilon)$ random variable to both, where $\varepsilon > 0$ is some small constant.