# A form of cumulative distribution

Let $f(x)$ and $g(x)$ be two probability density functions. Does the expression:

$$C = 2\int _{-\infty}^{\infty}\left[f(x)\int _{-\infty}^{x}g(y)\,dy\right]\,dx$$

have any meaningful graphical representation when $E[Y]>E[X]$, where $X$ has $pdf$ $f(x)$ and $Y$ has $pdf$ $g(x)$? Or, can it be expressed in a simpler fashion?

From some numerical simulations, it seems to approximately describe the “overlap” area of the two $pdf$’s. And if we let $g(x)==f(x)$ then $C$ is close to 1…
The overlap is defined as,
$$\int_{-\infty}^{\infty} \min(f(x),g(x)) dx$$

#### Solutions Collecting From Web of "A form of cumulative distribution"

If $X$ and $Y$ are independent continuous random variables with densities $f(x)$ and $g(y)$ then the probability that $Y$ is less than or equal to $X$ is

$$\Pr (Y \le X) = \int _{x=-\infty}^{\infty}[f(x)\int _{y=-\infty}^{x}g(y)dy]dx$$ and graphically is the probability of being below the line $y=x$.

$C$ is twice this. If $g(x)=f(x)$ for all $x$ then it is the probability of being above or below the line and (ignoring the zero measure of being on the line) is $1$.